![Introduction
The first impulse to study non-associative structures came in the first decades of the
20th century from the foundation of geometry, in particular from the investigation of
coordinate systems of non-Desarguesian planes. An additional interest forW. Blaschke
to treat loops and quasigroups systematically came from topological questions of dif-
ferential geometry, in particular from the topological behavior of geodesic foliations
[11]. R. Baer [7], A. A. Albert [4], [5], and R. H. Bruck [17] established the theory
of quasigroups and loops as an independent algebraic theory. For Baer the geometry
associated with a loop remains an important tool, for Bruck the theory of loops is a
part of general algebra [18]. Albert prefers to consider translations of a loop and to
see a loop as a section in the group generated by them. The development of the theory
of loops and quasigroups in the last 50 years was continued in the spirit of these three
directions. An important representative for the study of loops, quasigroups and their
associated geometry as abstract objects is V. D. Belousov [9]. The investigation of
loops within the framework of topological algebra, topological geometry and differ-
ential geometry gained importance by the work of A. I. Malcev [89], Κ. H. Hofmann
[47], H. Salzmann [120] and M. A. Akivis [3], The usefulness of analytic methods for
the theory of loops is shown in the work of L. V. Sabinin [116]. All these branches of
the theory of loops are collected and documented in [21].
Our aim here is to consider the theory of loops as a part of group theory; this means
to treat loops as sharply transitive sections in groups. Hence the group theoretical point
of view predominates the methods of non-associative algebra. We restrict our attention
to such classes of groups in which the simple objects are classified (e.g. finite groups,
algebraic groups, Lie groups). Since an essential part of our work will deal with
sharply transitive sections in Lie groups we shall use systematically also differential
geometric methods. From incidence structures we shall substantially use 3-nets which
are the geometries associated with loops; they are the most important tool for problems
concerning isotopism classes of loops. The local version of differentiable 3-nets are
3-webs which are coordinatized by local differentiable loops. Since the theory of
3-webs is systematically studied and the application of results on local loops can be
used in the global theory, the local point of view of 3-web geometry and the theory of
differentiable loops belong to the arsenal of our methods.
Binary operations " · " Μ χ Μ Μ on a set Μ with the property that for given
a, b e Μ the equations a · χ = b and y • a = b are uniquely solvable correspond to
sharply transitive subsets in transformation groups generated by these subsets. Indeed
the left transformations λα : χ ax are bijections and for given y, ζ e Μ there
exists precisely one left translation λα with Aa (j) — z. In Μ there is an element 1](https://image.slidesharecdn.com/88614-250427091143-1e337708/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-18-320.jpg)
![2 Introduction
for which 1-jc = jc-1 = jcif and only if the set of left translations contains the
identity. Sets with a binary operation " · " which correspond to sharply transitive sets
of transformations containing the identity are called loops.
If (L, ·) is a loop then the left translations χ ax (a e L) generate a permu-
tation group G on the set L. For non-associative L the group G is not sharply
transitive on Μ and the stabilizer Η of 1 e L in G is different from the identity.
The mapping σ : a λα : L —> G is & section with respect to the natural projec-
tion π : G G/Η. One may identify L with the factor space G/H and transport
the multiplication. Hence, the theory of loops coincides with the theory of triples
(G, Η, σ), where G is a group, Η a subgroup containing no normal non-trivial sub-
group of G and σ a section G/H —• G such that a(G/H) acts sharply transitively on
the left cosets xH, χ e G, and generates G. According to Baer [7], the set a ( G / H )
of representatives is sharply transitive on the factor set G/H if and only if σ (G/H) is
a set of representatives in G for every subgroup conjugate to Η. In [101] and [102] it
is stressed that it is always possible and fruitful to carry over the loop properties into
properties of the section σ.
If C denotes the category of topological spaces, differentiable manifolds, or of
algebraic varieties, then a loop is a C-loop if its multiplication, left division and right
division are C-morphisms. In the theory of triples (G, Η, σ), where G is a Lie group,
Η a closed subgroup and σ . G/H ^ G a C-morphism, the triple (G, Η,σ) defines
a C-loop multiplication by the rule
(χι H) • (X2H) = σ(χιΗ)χ2Η.
The first attempt to deal with differentiable and analytic loops was to follow the
ideas of Sophus Lie and to classify analytic loops by their tangential objects. In the
last 30 years this research program has been applied successfully to differentiable
Moufang loops (they are automatically analytic, cf. [102]) by Kuz'min, Kerdman and
Nagy. By their results the theory of differentiable Moufang loops has been carried
up to the level of the theory of Lie groups. Since the Hausdorff-Campbell formula
works also for binary Lie algebras, the theory of diassociative analytic local loops
may be treated successfully using binary Lie algebras the structure of which has been
determined by A. N. Grishkov. After this progress the attention turned to the class
of differentiable Bol loops. Here as well the investigations proceeded in the spirit of
Sophus Lie. With any analytic Bol loop L there is associated a Bol algebra B(L) such
that two Bol loops are locally isomorphic if and only if the corresponding Bol algebras
are isomorphic ([91], XII.8.12. Proposition). Hence the analytic local Bol loops can
be classified by the Bol algebras. But at this point a crucial difference from the theory
of Lie groups and Lie Moufang loops comes to light. Whereas any local Lie group or
local Lie Moufang loop may be embedded into a global one, this fact fails to be true for
local analytic Bol loops (and hence much more so for analytic local loops in general).
Already the classification of local analytic 2-dimensional Bol loops by Ivanov ([61],
[62], [63]) and the classification of global differentiable 2-dimensional loops show
the great difference between the varieties of differentiable global and local Bol loops.](https://image.slidesharecdn.com/88614-250427091143-1e337708/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-19-320.jpg)
![Introduction 3
Moreover, in Section 17 we exemplify by a class of analytic loops that the infinitesimal
behavior of a loop does not determine its global properties. Hence the investigation of
global differentiable loops cannot be reduced to that of local loops, and the procedure
to investigate suitable sections in Lie groups seems to us the only feasible method
for the classification of differentiable global loops for which the group topologically
generated by left translations is a Lie group. This restriction seems relatively mild to us
since differentiable loops with some weak associativity conditions have this property
(e.g. Bol loops, see [91], Proposition XII.2.14 and [102]).
With any loop L there is associated an incidence structure, called a 3-net Ν (cf. [8]);
if L is differentiable or analytic then Ν is also differentiable or analytic. Conversely,
to a 3-net Ν there corresponds a full class of isotopic loops, and with each point of Ν
taken as origin we may associate a coordinate loop defining the multiplication for the
points of a line through the origin graphically. Two coordinate loops of a net Ν are
isomorphic if and only if their points of origin are in the same orbit of the collineation
group Θ of T
V which preserves the directions ([8], p. 50).
Any identity in a loop corresponds to a configuration in the associated 3-net,
and configurations in 3-nets yield identities in some coordinate loops. The 3-nets
associated with Bol loops are of particular importance. For every line of a distinguished
pencil in such a net there exists an involutory collineation fixing this line pointwise.
These reflections generate a group Γ acting transitively on the distinguished pencil.
If the Bol loop is differentiable then Γ is a Lie group which induces the structure
of an affine symmetric space on the distinguished pencil. In an algebraic setting this
symmetric space is a left distributive groupoid which is called the core of the Bol loop.
These relations allow us to apply the rich theory of symmetric spaces as well as the
results on left distributive quasigroups in order to classify wide classes of Bol loops.
3-webs are incidence geometries associated with differentiable local loops; their
pencils of lines form 3 foliations. If a 3-web is associated with a differentiable local
Bol loop then for every leaf of one of the 3 foliations there exists a local reflection.
The group generated by these local reflections induces an affine locally symmetric
space on the manifold of leaves of this foliation; this connects the theory of local
differentiable Bol loops with the classical theory of locally symmetric spaces.
Our contribution has two parts. The first part contains the foundations of our new
methods and their applications to the extension theory of Bol loops, to the algebraic
theory of symmetric spaces and to the Lie theory of smooth loops. Moreover, we
classify strongly 2-divisible finite, differentiable and algebraic Bruck loops, and we
clarify the role of compactness of the group topologically generated by the left trans-
lations of a compact loop. In the second part we apply our methods to the topological
and differentiable loops on manifolds of small dimension.
In the Sections 1, 3 and 4 we develop the foundations for our point of view,
describe the interactions between the properties of sharply transitive sections and
the corresponding loops, discuss the relations between configurations in 3-nets and
corresponding identities in the coordinate loops and show what the local version of
these concepts is in the case of differentiable objects; at this instant the importance of](https://image.slidesharecdn.com/88614-250427091143-1e337708/85/Loops-in-Group-Theory-and-Lie-Theory-Peter-Nagy-20-320.jpg)
Loops in Group Theory and Lie Theory Péter Nagy
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- 5. Loops in Group Theory and Lie Theory Péter Nagy Digital Instant Download Author(s): Péter Nagy, Karl Strambach ISBN(s): 9783110170108, 3110170108 Edition: Reprint 2011 ed. File Details: PDF, 12.80 MB Year: 2002 Language: english
- 6. de Gruyter Expositions in Mathematics 35 Editors Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, Columbia University, New York R.O.Wells, Jr., Rice University, Houston
- 7. de Gruyter Expositions in Mathematics 1 The Analytical and Topological Theory of Semigroups, Κ. H. Hofmann, J. D. Lawson, J. S. Pym (Eds.) 2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues 3 The Stefan Problem, A. M. Meirmanov 4 Finite Soluble Groups, K. Doerk, T. O. Hawkes 5 The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin 6 Contact Geometry and Linear Differential Equations, V. E. Nazaikinskii, V. E. Shatalov, B. Yu. Sternin 7 Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, Μ. V. Zaicev 8 Nilpotent Groups and their Automorphisms, Ε. I. Khukhro 9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug 10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini 11 Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon, G. Zhao 12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub 13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Nazarov, B. A. Plamenevsky 14 Subgroup Lattices of Groups, R. Schmidt 15 Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, P. H. Tiep 16 The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti, A. J. Sommese 17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno 18 Unitary Representation Theory of Exponential Lie Groups, H. Leptin, J. Ludwig 19 Blow-up in Quasilinear Parabolic Equations, A. A. Samarskii, V.A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov 20 Semigroups in Algebra, Geometry and Analysis, Κ. H. Hofmann, J. D. Lawson, Ε. B. Vinberg (Eds.) 21 Compact Projective Planes, H. Salzmann, D. Betten, Τ. Grundhöf er, Η. Hühl, R. Löwen, M. Stroppel 22 An Introduction to Lorentz Surfaces, Τ. Weinstein 23 Lectures in Real Geometry, F. Broglia (Ed.) 24 Evolution Equations and Lagrangian Coordinates, A. M. Meirmanov, V. V. Pukhnachov, S. I. Shmarev 25 Character Theory of Finite Groups, B. Huppert 26 Positivity in Lie Theory: Open Problems, J. Hilgert, J. D. Lawson, K.-H. Neeb, Ε. B. Vinberg (Eds.) 27 Algebra in the Stone-Cech Compactification, N. Hindman, D. Strauss 28 Holomorphy and Convexity in Lie Theory, K.-H. Neeb 29 Monoids, Acts and Categories, M. Kilp, U. Knauer, Α. V. Mikhalev 30 Relative Homological Algebra, Edgar E. Enochs, Overtoun M. G. Jenda 31 Nonlinear Wave Equations Perturbed by Viscous Terms, Viktor P. Maslov, Petr P. Mosolov 32 Conformal Geometry of Discrete Groups and Manifolds, Boris N. Apanasov 33 Compositions of Quadratic Forms, Daniel B. Shapiro 34 Extension of Holomorphic Functions, Marek Jarnicki, Peter Pflug
- 8. Loops in Group Theory and Lie Theory by Peter Τ. Nagy Karl Strambach W DE _G Walter de Gruyter · Berlin · New York 2002
- 9. Authors Peter Τ. Nagy Institute of Mathematics University of Debrecen P.O.B. 12 4010 Debrecen, Hungary nagypeti@math.klte.hu Karl Strambach Mathematisches Institut der Universität Erlangen-Nürnberg Bismarckstr. Υ/τ 91054 Erlangen, Germany strambach@mi. uni-erlangen. de Mathematics Subject Classification 2000: 22-02, 20-02; 20N05, 20G20, 22E60, 51A20, 51A25, 51H20, 53C30, 53C35, 57S10, 57S15, 57S20 Key words: Loops, quasigroups, Lie groups, Lie transformation groups, Lie algebras, tangent alge- bra, symmetric spaces, 3-net, 3-web, configurations, collineation groups, Bol loops, Bruck loops, Moufang loops, topological translation planes, quasifields © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress — Cataloging-in-Publication Data Nagy, Peter Tibor. Loops in group theory and Lie theory / Peter T. Nagy, Karl Stram- bach. p. cm ISBN 3-11-017010-8 1. Loops (Group theory) 2. Lie groups. I. Strambach, Karl. II. Title. QA 174.2 .N34 2001 512'.2-dc21 2001042383 Die Deutsche Bibliothek — Cataloging-in-Publication Data Nagy, Peter T.: Loops in group theory and Lie theory / by Peter T. Nagy ; Karl Stram- bach. - Berlin ; New York : de Gruyter, 2002 (De Gruyter expositions in mathematics ; 35) ISBN 3-11-017010-8 © Copyright 2002 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Typesetting using the authors' TgX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen. Cover design: Thomas Bonnie, Hamburg.
- 10. Preface In this book the theory of loops is viewed as a part of group theory. Special attention is paid to topological, differentiable, and algebraic loops which are treated using Lie groups, algebraic groups and symmetric spaces. The topic of this book requires some patience with computations in non-associative structures; the basic notions needed as well as facts from the general theory of loops are collected in Section 1.1. The reader is expected to have a basic knowledge of group theory and some fa- miliarity with Lie groups, Lie algebras and differential geometry of homogeneous spaces. For the new material complete proofs are given. For the needed known results, the proofs of which require extensive preparation not concerning our topic, precise available references are quoted. We tried to organize the sections of this book in such a way that they are readable mostly independently from each other. But e.g. the sections 5,6, 8,9,10,11 and 13 as well as 2, 9, 10, 14, 15 and 16 treat closely related topics from the same point of view and with similar methods. The second part of this book is mainly devoted to examine the possibilities in small dimensions; the variety of constructed examples shows that a classification is feasible only for loops with some weak associativity condition. The authors are grateful to the Volkswagen-Stiftung (RiP-Program at Oberwol- fach), to the Paul Erdös Summer Research Center for Mathematics, to Deutscher Akademischer Austauschdienst and to the Hungarian Ministry of Education for the partial support of this project. Special thanks are due to A. Figula, T. Grundhöfer, Ο. Η. Kegel, G. P. Nagy, A. Schleiermacher and V. Zambelli who read the whole or some parts of this manuscript and gave us many useful suggestions. We also want to mention a workshop at the Palacky University of Olomouc supported by GACR 201/99/0265 of J. Mikes. Finally we thank L. Kozma for the preparation of the book in final form.
- 12. Notation In principle, we try to avoid in the text symbols and abbreviations, in particular if they are not of standard use in the literature. As usual Z, R, C denote the integer, real and complex numbers, respectively. H means the (real) quaternions and Ο the (real) octonions (Cayley numbers). If (Μ, ·) is a loop then χ · (y · z) is often denoted by λ: · yz- If G is a group and Μ is a subset of G then (M) is the subgroup generated by M. If Μ is a subset in a topological space S then Μ is the closure of Μ in S. If Η is a subgroup of G then we write Η < G and G/H is the set of left cosets of Η in G. G' means mainly the commutator subgroup of G. For the classical groups we use the standard notation as e.g. GL„(^T), PGL„(A"), SOn{K), PSL2 (/0, PSpn (K), PSU„(/0 for a suitable field K. For the Lie algebras of the classical groups we use the corresponding notation gln, $on, on, $un, etc. If the algebraic structures M and M2 are isomorphic we write Mi = M2. If φ : A —> ß is a map then the image of an element χ e A is usually denoted by φ(χ); but sometimes we write χ φ or χφ. For structures M and Mi the structure M χ Mi is given as usual by {C*i,*2);e M,x2 e M2}. Let Gi and G2 be groups and φ : G — G 2 be a homomorphism. Then we distinguish between the following two subgroups of G χ Gi'· Gl X <p(Gi) = {{χ,φ{χΐ)) Xl,X2 € G l ) and (ΰι,φ(ΰι)) = {(χ,φ(χ)); χ G Gj} . Ν χ Κ denotes a semidirect product of the normal subgroup Ν by the group Κ. For two sets A and Β the relation A c Β means that Λ is a (not necessarily proper) subset of Β. The differentiability class Cr (r — 1 , . . . , 00) is the class of r-times continuously differentiable maps and manifolds, Cw denotes the class of real analytic maps and manifolds.
- 14. Contents Preface ν Notation vii Introduction 1 Parti General theory of transitive sections in groups and the geometry of loops 1 Elements of the theory of loops 13 1.1 Basic facts on loops 13 1.2 Loops as sections in groups 17 1.3 Topological loops and differentiable loops 29 2 Scheerer extensions of loops 42 3 Nets associated with loops 53 4 Local 3-nets 60 5 Loop-sections covered by 1-parameter subgroups and geodesic loops 65 6 Bol loops and symmetric spaces 80 7 Bol nets 95 8 Strongly topological and analytic Bol loops 100 9 Core of a Bol loop and Bruck loops 102 9.1 Core of a Bol loop 102 9.2 Symmetric spaces on differentiable Bol loops 113 10 Bruck loops and symmetric quasigroups over groups 120
- 15. χ Contents 11 Topological and differentiable Bruck loops 129 12 Bruck loops in algebraic groups 143 13 Core-related Bol loops 150 14 Products and loops as sections in compact Lie groups 166 14.1 Pseudo-direct products 166 14.2 Crossed direct products 168 14.3 Non-classical differentiable sections in compact Lie groups 170 14.4 Differentiable local Bol loops as local sections in compact Lie groups 173 15 Loops on symmetric spaces of groups 174 15.1 Basic constructions 174 15.2 A fundamental reduction 182 15.3 Core loops of direct products of groups 186 15.4 Scheerer extensions of groups by core loops 190 16 Loops with compact translation groups and compact Bol loops 194 17 Sharply transitive normal subgroups 208 Part II Smooth loops on low dimensional manifolds 18 Loops on 1-manifolds 235 19 Topological loops on 2-dimensional manifolds 249 20 Topological loops on tori 256 21 Topological loops on the cylinder and on the plane 262 21.1 2-dimensional topological loops on the cylinder 262 21.2 Non-solvable left translation groups 264 22 The hyperbolic plane loop and its isotopism class 276 23 3-dimensional solvable left translation groups 289 23.1 The loops L(a) and their automorphism groups 290 23.2 Sharply transitive sections in £2 x K. 298 23.3 Sections in the 3-dimensional non-abelian nilpotent Lie group 308 23.4 Non-existence of strongly left alternative loops 312
- 16. Contents xi 24 4-dimensional left translation group 317 25 Classification of differentiable 2-dimensional Bol loops 321 26 Collineation groups of 4-dimensional Bol nets 329 27 Strongly left alternative plane left A-loops 335 28 Loops with Lie group of all translations 338 29 Multiplicative loops of locally compact connected quasifields 344 29.1 2-dimensional locally compact quasifields 345 29.2 Rees algebras Qe 346 29.3 Mutations of classical compact Moufang loops 348 Bibliography 351 Index 359
- 18. Introduction The first impulse to study non-associative structures came in the first decades of the 20th century from the foundation of geometry, in particular from the investigation of coordinate systems of non-Desarguesian planes. An additional interest forW. Blaschke to treat loops and quasigroups systematically came from topological questions of dif- ferential geometry, in particular from the topological behavior of geodesic foliations [11]. R. Baer [7], A. A. Albert [4], [5], and R. H. Bruck [17] established the theory of quasigroups and loops as an independent algebraic theory. For Baer the geometry associated with a loop remains an important tool, for Bruck the theory of loops is a part of general algebra [18]. Albert prefers to consider translations of a loop and to see a loop as a section in the group generated by them. The development of the theory of loops and quasigroups in the last 50 years was continued in the spirit of these three directions. An important representative for the study of loops, quasigroups and their associated geometry as abstract objects is V. D. Belousov [9]. The investigation of loops within the framework of topological algebra, topological geometry and differ- ential geometry gained importance by the work of A. I. Malcev [89], Κ. H. Hofmann [47], H. Salzmann [120] and M. A. Akivis [3], The usefulness of analytic methods for the theory of loops is shown in the work of L. V. Sabinin [116]. All these branches of the theory of loops are collected and documented in [21]. Our aim here is to consider the theory of loops as a part of group theory; this means to treat loops as sharply transitive sections in groups. Hence the group theoretical point of view predominates the methods of non-associative algebra. We restrict our attention to such classes of groups in which the simple objects are classified (e.g. finite groups, algebraic groups, Lie groups). Since an essential part of our work will deal with sharply transitive sections in Lie groups we shall use systematically also differential geometric methods. From incidence structures we shall substantially use 3-nets which are the geometries associated with loops; they are the most important tool for problems concerning isotopism classes of loops. The local version of differentiable 3-nets are 3-webs which are coordinatized by local differentiable loops. Since the theory of 3-webs is systematically studied and the application of results on local loops can be used in the global theory, the local point of view of 3-web geometry and the theory of differentiable loops belong to the arsenal of our methods. Binary operations " · " Μ χ Μ Μ on a set Μ with the property that for given a, b e Μ the equations a · χ = b and y • a = b are uniquely solvable correspond to sharply transitive subsets in transformation groups generated by these subsets. Indeed the left transformations λα : χ ax are bijections and for given y, ζ e Μ there exists precisely one left translation λα with Aa (j) — z. In Μ there is an element 1
- 19. 2 Introduction for which 1-jc = jc-1 = jcif and only if the set of left translations contains the identity. Sets with a binary operation " · " which correspond to sharply transitive sets of transformations containing the identity are called loops. If (L, ·) is a loop then the left translations χ ax (a e L) generate a permu- tation group G on the set L. For non-associative L the group G is not sharply transitive on Μ and the stabilizer Η of 1 e L in G is different from the identity. The mapping σ : a λα : L —> G is & section with respect to the natural projec- tion π : G G/Η. One may identify L with the factor space G/H and transport the multiplication. Hence, the theory of loops coincides with the theory of triples (G, Η, σ), where G is a group, Η a subgroup containing no normal non-trivial sub- group of G and σ a section G/H —• G such that a(G/H) acts sharply transitively on the left cosets xH, χ e G, and generates G. According to Baer [7], the set a ( G / H ) of representatives is sharply transitive on the factor set G/H if and only if σ (G/H) is a set of representatives in G for every subgroup conjugate to Η. In [101] and [102] it is stressed that it is always possible and fruitful to carry over the loop properties into properties of the section σ. If C denotes the category of topological spaces, differentiable manifolds, or of algebraic varieties, then a loop is a C-loop if its multiplication, left division and right division are C-morphisms. In the theory of triples (G, Η, σ), where G is a Lie group, Η a closed subgroup and σ . G/H ^ G a C-morphism, the triple (G, Η,σ) defines a C-loop multiplication by the rule (χι H) • (X2H) = σ(χιΗ)χ2Η. The first attempt to deal with differentiable and analytic loops was to follow the ideas of Sophus Lie and to classify analytic loops by their tangential objects. In the last 30 years this research program has been applied successfully to differentiable Moufang loops (they are automatically analytic, cf. [102]) by Kuz'min, Kerdman and Nagy. By their results the theory of differentiable Moufang loops has been carried up to the level of the theory of Lie groups. Since the Hausdorff-Campbell formula works also for binary Lie algebras, the theory of diassociative analytic local loops may be treated successfully using binary Lie algebras the structure of which has been determined by A. N. Grishkov. After this progress the attention turned to the class of differentiable Bol loops. Here as well the investigations proceeded in the spirit of Sophus Lie. With any analytic Bol loop L there is associated a Bol algebra B(L) such that two Bol loops are locally isomorphic if and only if the corresponding Bol algebras are isomorphic ([91], XII.8.12. Proposition). Hence the analytic local Bol loops can be classified by the Bol algebras. But at this point a crucial difference from the theory of Lie groups and Lie Moufang loops comes to light. Whereas any local Lie group or local Lie Moufang loop may be embedded into a global one, this fact fails to be true for local analytic Bol loops (and hence much more so for analytic local loops in general). Already the classification of local analytic 2-dimensional Bol loops by Ivanov ([61], [62], [63]) and the classification of global differentiable 2-dimensional loops show the great difference between the varieties of differentiable global and local Bol loops.
- 20. Introduction 3 Moreover, in Section 17 we exemplify by a class of analytic loops that the infinitesimal behavior of a loop does not determine its global properties. Hence the investigation of global differentiable loops cannot be reduced to that of local loops, and the procedure to investigate suitable sections in Lie groups seems to us the only feasible method for the classification of differentiable global loops for which the group topologically generated by left translations is a Lie group. This restriction seems relatively mild to us since differentiable loops with some weak associativity conditions have this property (e.g. Bol loops, see [91], Proposition XII.2.14 and [102]). With any loop L there is associated an incidence structure, called a 3-net Ν (cf. [8]); if L is differentiable or analytic then Ν is also differentiable or analytic. Conversely, to a 3-net Ν there corresponds a full class of isotopic loops, and with each point of Ν taken as origin we may associate a coordinate loop defining the multiplication for the points of a line through the origin graphically. Two coordinate loops of a net Ν are isomorphic if and only if their points of origin are in the same orbit of the collineation group Θ of T V which preserves the directions ([8], p. 50). Any identity in a loop corresponds to a configuration in the associated 3-net, and configurations in 3-nets yield identities in some coordinate loops. The 3-nets associated with Bol loops are of particular importance. For every line of a distinguished pencil in such a net there exists an involutory collineation fixing this line pointwise. These reflections generate a group Γ acting transitively on the distinguished pencil. If the Bol loop is differentiable then Γ is a Lie group which induces the structure of an affine symmetric space on the distinguished pencil. In an algebraic setting this symmetric space is a left distributive groupoid which is called the core of the Bol loop. These relations allow us to apply the rich theory of symmetric spaces as well as the results on left distributive quasigroups in order to classify wide classes of Bol loops. 3-webs are incidence geometries associated with differentiable local loops; their pencils of lines form 3 foliations. If a 3-web is associated with a differentiable local Bol loop then for every leaf of one of the 3 foliations there exists a local reflection. The group generated by these local reflections induces an affine locally symmetric space on the manifold of leaves of this foliation; this connects the theory of local differentiable Bol loops with the classical theory of locally symmetric spaces. Our contribution has two parts. The first part contains the foundations of our new methods and their applications to the extension theory of Bol loops, to the algebraic theory of symmetric spaces and to the Lie theory of smooth loops. Moreover, we classify strongly 2-divisible finite, differentiable and algebraic Bruck loops, and we clarify the role of compactness of the group topologically generated by the left trans- lations of a compact loop. In the second part we apply our methods to the topological and differentiable loops on manifolds of small dimension. In the Sections 1, 3 and 4 we develop the foundations for our point of view, describe the interactions between the properties of sharply transitive sections and the corresponding loops, discuss the relations between configurations in 3-nets and corresponding identities in the coordinate loops and show what the local version of these concepts is in the case of differentiable objects; at this instant the importance of
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- 22. CHAPTER XIII. A RACE IN THE AIR. “Don’t read any more chapters from your dream-book!” warned Ben. “We’ve materialized the coat, the whiskers, the dickey, the wing collar, the red tie, and the felt hat Colleton wore away from his office that day, and I think that’s about enough!” “Materialized ’em through three thousand miles of space, at that!” laughed Jimmie. “If we could materialize Colleton as easily, we might have a little time for hunting on this trip.” The aeroplane which had been reported by Carl was still quite a distance to the west. It carried a light which appeared not much larger than a good-sized planet from where the boys stood. The hum of the motors sounded faintly from the distance. “It’s pears to potatoes,” exclaimed Carl, “that she’s going up to that old camp!” “If she does, she’ll find a man drunk in the cavern, and that’s all!” “And a lot of whiskey and brandy!” suggested Jimmie. The aeroplane moved slowly to the north and west, and presently the boys were able to see something more than the dancing light. “She’s going to the old camp all right!” Ben announced, after looking at the machine through his field-glass for some minutes. “At least, she is headed in that direction now.” “And why shouldn’t she be going to the camp?” asked the Englishman. “Because only two classes of people are now much interested in that locality!” cried Ben. “The class most interested is the criminal class. The other is the official class. I have a notion that the
- 23. criminals are pretty well disposed of to-night,” the boy continued, “and it isn’t time for the officers to return. Besides,” he went on, “they wouldn’t be apt to return in an aeroplane.” “I’ll tell you how we can soon find out all about it!” suggested Jimmie. “I know how we can find out all about that machine!” “No, you don’t,” laughed Mr. Havens. “You don’t get away in any machine to-night! It spells trouble when you get away after dark!” “Je-rusalem!” exclaimed Jimmie, in a disgusted tone. “I might have known I’d need my knitting when I came out on this trip! If I listened to all the advice I get from you fellows, I’d sit down here and knit myself a pair of socks, or a cream-colored necktie, just like a perfect little lady. What’s the matter with a game of checkers? Wouldn’t that be too exciting for you?” he added, with a grin. “I don’t think there’s been any lack of excitement up to date,” laughed Mr. Havens. “Say,” Ben exclaimed, directly, “we really ought to go and see what that Crooked Terry is doing. You know I set out once to get a duplicate copy of the map of this country which he is supposed to carry in his head.” “Is this a conspiracy to get away from camp again?” demanded the millionaire. “Do you want to leave me here alone all the time?” “We’ll leave Carl and Mr. Claude Mercer Du Bois to keep you company,” suggested Jimmie. “If you don’t mind,” the Englishman cut in, “I’d like to have me dinner now, don’t you know.” “I’ll bet it’s all scorched to coals!” cried Carl, rushing to the fire. In a moment he called back that the ham and eggs and coffee were just as they should be, and the Englishman was soon eating heartily. The strange aeroplane was still in sight. In fact a great deal closer than when it had first been discovered. It was now over the center of the valley, still pointing toward the shelf from which the signals had been given the night before. While the boys watched and waited, undecided as to the correct course to pursue, the machine passed over the snow-tipped summit and disappeared.
- 24. “Some aviator out for a view of the mountains, probably,” Mr. Havens suggested. “He seems to be keeping on his way pretty well.” “I’ve got a hunch,” Jimmie insisted, “that that aeroplane has something to do with this Kuro case!” “Aw, cut out the dream-book!” advised Ben. “Didn’t my dreams come true?” demanded the boy. “You’ll have to show me!” declared Carl. “Don’t you suppose there’s more than one false beard, more than one sporty coat, and more than one dickey with wing collars and a red necktie in the world?” Jimmie ran out to the Louise, showing by his manner that he considered the question too trivial to be answered. “Come on, Ben,” he called. “We’ll go up high enough to see where that aeroplane went. If she’s still on her way east, we’ll come down and go to bed, like good little boys. If she’s hovering around the other side of the summit, we’ll catch the aviator and put him through the third degree. We’ll have a good ride, anyway!” No further objections were offered, and the Louise was soon in the air. The boys kept her down so that her lights could not be seen from the other side of the ridge until they came to the vicinity of the gully, then they lifted suddenly and crossed the summit, shivering in the icy air of the mountaintop. The aeroplane lay just below on the ledge which had been occupied by the Louise on the previous night. Three lights were in sight. The lamp on the forward framework of the machine was burning brightly, and two men were walking along the ledge with electric searchlights in their hands. They did not appear to be surprised at the appearance of the Louise. “I wonder what they’ve lost,” said Jimmie, his teeth fairly chattering with the cold. “Suppose we go down and ask.” Ben circled the Louise into the warmer air of the valley on the other side of the summit, and then moved slowly to the west. As he did so, the strange aeroplane leaped into the air and darted off to the south. She seemed to be a speedy machine, for she swept away from the Louise with wonderful ease.
- 25. “You just wait till I get turned around and get the motors on,” Ben muttered, “and I’ll show you that we can go some!” The stranger was some distance in the lead before the Louise was well under way. After that it seemed to the boys that they gained, although very slowly. The machines both kept as low down as possible and ran to the full power of their motors. The rush of wind and the clatter of the motors effectually checked verbal communication, but Jimmie pointed significantly to the machine ahead and then nodded determinedly. “Let her go,” muttered Jimmie under his breath. “We had a race something like this in Old Mexico, and the other machine brought up in the Pacific ocean. That was a race that ought to have been written up!” In the meantime, those watching from the camp saw the strange aeroplane dart swiftly over the ridge and head into the succession of valleys running to the west of the range. A few moments later she was followed by the Louise. “I’d like to know what those crazy boys are doing!” exclaimed Mr. Havens, rather impatiently. “They’re trying to catch that machine!” laughed Carl. “But why should they take the chance of an accident by running at such speed in the night-time?” asked the millionaire. “There are holes in the air just as there are holes in the surface of the earth, and the first thing they know they’ll drop down about a thousand feet and tip over! It’s a risky proposition!” “That’s what it is!” returned Carl shaking his head gravely. “It’s a risky proposition, and if you say the word I’ll jump on the Ann and go and tell them to come back!” The aviator laughed at the innocent manner of the boy, and the Englishman regarded the two with a stare of wonder. “I never saw anything like it, don’t you know!” the latter said. “You’re likely to see something like it several times before you get out of the mountains!” laughed Carl. “Say, Mr. Havens,” the boy went on, “we don’t want that strange machine to come here and beat us in a race, do we? I don’t think the Louise is making much of a show,
- 26. and so, if you don’t mind, I’ll take out the Ann and run ’em both down. It would be a lovely race!” “I wouldn’t mind going with the lad, don’t you know!” exclaimed DuBois showing great excitement. “If you do go,” replied Mr. Havens, “you’ll get fined a year’s salary if you don’t catch both machines!” “Oh, I’ll catch ’em all right!” Carl exclaimed. “The Ann can run around both those old ice wagons, and then have plenty of time to spare!” “The Ann can beat any aeroplane that was ever built!” replied Mr. Havens. “She was built for a record-breaker.” To tell the truth, the aviator was not exactly pleased at the idea of remaining alone in the camp while the two engaged in the race, but the sporting strain was strong in the man’s blood, and he was proud of his matchless machine, so he consented, principally because he wanted the Ann to win in a race which promised to be a hot one! “I wish the other machines would keep in sight so I could watch the struggle,” he said as Carl sprang toward the Ann. “Do you know,” the Englishman observed, “I rather like the spirit of the lad!” “He’s all right,” replied the millionaire. “But,” he added, “I didn’t think you had the courage to get into such a game.” “To tell you the truth,” DuBois replied, “I was tolerably well frightened during my ride here, but I think I can now trust myself in any place that lad is willing to go.” Mr. Havens saw the Ann rise swiftly into the air; rise to a height which must have chilled the blood of those on board, and then flash off to the south. The two aeroplanes were still in view although their lights showed dimly. From his position in the tent the aviator could not determine whether or not the Louise was gaining. He saw that the great light of the Ann was rapidly closing the gap between the nearest lamp and herself, and had no doubt of the outcome of the race. While he gazed one of the lights ahead dropped. Without knowing which machine had fallen, he crept to a corner of the tent on his hands and knees and brought out a night glass.
- 27. CHAPTER XIV. THE END OF THE FLIGHT. When the Ann rose above the valley Carl saw the Louise some distance to the south. The strange machine was still in the lead, but the boys appeared to be gaining on her. Both were going fast. The sky was now tolerably clear, although a brisk wind driving in from the west was bringing fleecy clouds from the Pacific coast. There would be a moon sometime between midnight and morning, but the prospects were that there would be a bank of driving clouds stretched over the earth before she showed herself. The Englishman, unfamiliar with aeroplaning, began asking questions of the boy as soon as they were in the air, but, as the racing of the motors and the rush of the air drowned his voice, he soon lapsed into silence and contented himself with such views of the distant summit as he could secure. Several times he flung out an arm—including the shining stars, the drifting clouds, the wide stretch of mountain and valley in the sweep of it—and Carl understood that he was saying in the only language available there how much he loved the wild beauty and the majesty of it all. After a time the strange aeroplane began to seek the higher levels. She climbed up, up, up until the summit showed white and sparkling under her flying planes. Carl saw the Louise following the stranger into the snow zone and wondered at it. To the boy it seemed that the distance traveled upward might better be gained in level flight. Every unnecessary foot of altitude seemed to him to be a foot lost in the race.
- 28. “Ben doesn’t have to follow the stranger in the air,” he mused as he shot the Ann ahead on the same level he had been traveling. “All he has to do in order to overtake her is to keep her in sight and go faster than she does. He lost several yards by following her up to the summit.” After a time the stranger changed her tactics turning to the west and seeking the valley again. The Louise followed in her wake as before and seemed to be gaining. The Ann was traveling much faster than either of the others and would soon be within striking distance. That was a mad race under the stars. The stranger seemed to develop new speed possibilities as she swept along. The Louise appeared to be losing ground. The Ann swept forward relentlessly and was soon close to the rear machine. Then a remarkable thing happened. The aeroplane in advance dropped like a plummet. It seemed to Carl, watching her light eagerly from his seat on the Ann, that she ceased her forward motion and lost her buoyancy at the same moment. He could not, of course, see the bulk of the machine but he could see her light. The light seemed to be down to the surface of the earth in a minute. The Louise, following on, dropped, too. To the watching boy the falling of the two aeroplanes seemed as if they had dropped over a precipice. Although not a very old or experienced aviator, Carl sensed what had taken place. The machines had dropped into a hole in the air! As is well-known to those conversant with the navigation of the air, there are actually “holes” in the atmosphere—holes into which machines drop as they would drop into a pit on the surface of the earth. There are also cross currents which tug at the planes in a wholly unaccountable manner. These holes in the air result, of course, from conditions of temperature. They are dreaded by all aviators, and one of the first things taught in schools of aviation is to keep such control of his machine as will enable him to handle her successfully when such pitfalls and cross currents are encountered.
- 29. Carl had learned this lesson well under the tuition of Mr. Havens, and his first act when the Louise fell was to shift the Ann far away to the north of the place of descent. He dropped down, too, in a moment in order to see what had happened to the other machines. The stranger lay a wreck in a rocky valley below and the Louise, some distance in the rear, was fluttering down. It seemed to Carl that some of her guy wires had been broken during the strain of the fall, and that she was almost beyond control of his chum. Circling about the wrecked machine and the one which appeared to be in danger of being wrecked, Carl dropped lower and lower until at last his light disclosed a level bank at the side of a stream where he believed a landing might be effected. By this time the Louise lay on the ground. He could not tell whether she had fallen with a crash or had gradually settled down. However, her lights were still burning, and he could see one of the boys moving about. The lights of the other machine were out. The Ann came very near tipping over into the stream as Carl landed and a growth of bushes at the water’s edge scraped the ends of the planes cruelly as she settled down. Without stopping to inspect any damage that might have been done to the aeroplane, Carl dashed over to the Louise. The boys were at that moment leaving their machine, turning their footsteps in the direction of the stranger. It was quite dark in the valley, as the timber line extended far up on the easy slope, and the boys were using their electrics as they moved along. “Are you boys all right?” asked Carl, as he came panting to their side. “I thought I heard one of you groaning!” “We’re all right!” exclaimed Jimmie. “The Louise strained her guy wires when we struck that hole in the air, but we managed to flutter down. Except for the broken guy wires the machine is as good as ever she was. We can fix the guy wires right here!” “But the other machine fell!” Ben added. “When she went into the hole the driver wasn’t attending to his business, so she twisted sideways and turned turtle a hundred feet from the ground. We’re going over there now to see if the man is dead.”
- 30. “This ends my after-dark journeys in the air!” declared Carl. “There’s no sense in it!” added Ben. DuBois, the Englishman, now came stumbling through the darkness and paused in the circle of light made by the electrics. He was still shivering with cold, although the Ann had not mounted to a high level. “What’s the bloody trouble?” he asked. “You’re right about the trouble being a bloody one!” Jimmie replied. “The man we were chasing wrecked his machine.” DuBois looked the Louise over critically. “This one fell, too, don’t you know,” he said. “Oh, we always come down like that!” declared Jimmie. The Englishman stood leaning against the Louise when the boys left for the wrecked machine. It was all new to him, but he seemed to be taking in the situation slowly. When the boys reached the wreck the aviator who had driven the machine lay on the ground, a dozen or more feet away from the seat he had occupied. He appeared to be quite dead. The body had the appearance of having fallen free of the machine some distance up in the air and crushed down upon the soft grass of the valley. Ben stooped over the still figure for a moment and then turned to his chums with a queer look on his face. “Do you remember the heavy man in brown who stood in the corridor at the door of Colleton’s room?” he asked. “We certainly do!” answered Jimmie. “I’ve been thinking about that husky man in brown ever since Mr. Havens told us the story.” “What brings that to your mind now?” asked Carl. “Look at this body!” answered Ben. “Look at the heavily-bearded face. Look at the brown suit. Look at the refined and yet business- like makeup of the man. Even in death he seems domineering and forceful.” “That man was no aviator!” Jimmie exclaimed. “His handling of the machine showed that!” Carl put in. “And do you think?” asked Jimmie in a moment, “that——” The boy was interrupted by the sudden appearance of the Englishman, who came out of the darkness with his hands pushed
- 31. far into his pockets and his teeth rattling with the cold. The boys stepped aside as he drew near the body on the ground and waited for him to speak. “Don’t you remember,” Jimmie whispered to Ben, “that DuBois bought that hand-bag of a porter on the Pullman-car which carried a sick man in a private stateroom across the continent?” “What’s that got to do with it?” demanded Carl. “Wait a moment!” advised Jimmie. “Watch the Englishman’s face to see if he recognizes the dead man.” “Is this another page out of your dream-book?” asked Carl. “How do we know” demanded Jimmie impatiently, “that DuBois didn’t see a score of times on that trip the man who occupied the stateroom with the man who was sick?” “Oh, I see!” Carl said. “You think this man lying here dead is the man who stood at the corridor door that day?” “I didn’t say so!” whispered Jimmie. “I said to watch for some sign of recognition in the Englishman’s face.” The Englishman bent over the dead man, searching outline of face and figure under the dim light of the stars. The boys heard a little exclamation of impatience, and then DuBois motioned to Ben to advance his searchlight so as to bring the dead face under its rays. Ben did so immediately and the Englishman stood for what seemed to be a long time looking downward with a puzzled face. He brought his hand to his brow several times as if seeking to urge his slow brain into action and finally turned away without saying a word. “That was a bad fall!” Ben said, seeking to engage the Englishman in conversation. “We came near lying where he does this minute.” “A bad fall!” repeated the Englishman. “Do you know who the man is?” “Never saw him before to-night!” replied Ben. “You might look in his pockets, don’t you know!” suggested DuBois. “That’s a good suggestion!” cried Jimmie who had been listening to the conversation. “I’ll see what I can find right now, if you’ll hold the light, Carl,” he added.
- 32. Carl advanced with the light and a thorough search was made of the dead man’s clothing. The pockets were entirely empty save for a watch, a pocket-knife, a fountain pen and a collapsible tube of adhesive material. The underclothing, shirt, collar and cuffs were new and bore no name or laundry mark. The collar of the coat bore the trade mark of a well-known firm of manufacturers dealing only in ready-to-wear clothing. On the inside of the right sleeve was the union label of the garment workers. The serial number of the label was blurred and could not be read. Ben opened the watch case eagerly but found no initials on the inside. There was nothing whatever about the man to give information as to his name, occupation, or place of residence. That he had been a business man and not a professional aviator was clear to the boys but their information went no farther. The Englishman stood by while the articles taken from the dead man accumulated on the grass but said nothing. Now and then he stepped closer and looked down into the white face. “Don’t you know,” he said presently, “I think I’ve seen that man before!” Jimmie nudged Carl impulsively but said nothing. “You might have seen him in Washington,” suggested Ben. “No,” answered the Englishman. “The man is not associated in my mind with anything that took place in your capital city.” “On the boat coming over?” suggested Carl with a wink at Jimmie. “No-o,” hesitated the Englishman. “I can’t associate that face with anything on board the steamer. It might have been on the train coming across the continent,” he went on in a musing tone. “It might have been in the Pullman on the way over.” “If your recollection is so indistinct,” Jimmie put in, “it must be because you didn’t see much of him on the train. Perhaps he remained in his stateroom most of the time.” “That’s clever of you, don’t you know!” the Englishman drawled. “Your suggestion of the stateroom brings it all back. This dead man, don’t you know, often passed in and out of the stateroom door and we noticed his goings and comings because he never permitted any one to see inside the door, don’t you know.”
- 33. “Did the man lose anything on the train?” asked Jimmie. “Yes, he told the porter he had lost his bag.”
- 34. CHAPTER XV. THE MAN IN THE STATEROOM. “Did he make much of a row about it?” asked Jimmie. “No,” was the answer, “because the porter convinced him that it had accidentally fallen from the vestibule during a short stop in one of the passes. The fellow seemed glad to know that it was gone!” “How could it get lost from the vestibule?” “The fellow admitted leaving it somewhere outside the stateroom after taking it to the toilet with him.” “Did it ever occur to you,” asked Jimmie, “that you bought the hand-bag the porter stole from the man lying here dead?” “That’s a queer suggestion, don’t you know!” said the Englishman. “Well, how did the porter come to have the bag to sell if he hadn’t picked it up somewhere on the train?” “That’s a clever question!” asserted the Englishman. “But look here,” he went on, “why should a man like this one have a false shirt front and a false beard in his luggage?” “I think I could tell you why if I tried very hard,” answered Jimmie, “but we’d better pass that up for the present.” “Yes,” Ben said, “I think we’d better give this man decent burial, repair the Louise as far as possible, and start back to camp.” “I don’t see how we’re going to open a grave,” Carl said. “We can make a shallow one, I guess,” Ben answered, “and then use plenty of stones for covering. Of course we’ll notify the mounted police as soon as we get to a station, and they will undoubtedly take the body out. Somewhere, undoubtedly, this man had relatives and friends, and they ought to know the manner of his death.”
- 35. It was not very difficult making a shallow grave in the soft soil, although the boys had no suitable tools to work with. When at last the body was wrapped in a canvas shroud, composed of material taken from the planes of the wrecked machine, and laid into the grave it was covered to a considerable height with heavy rocks taken from the slope. This task completed, the boys took guy wires from the now useless aeroplane and repaired the breakage on the Louise. The tanks of the Louise being about half empty, the gasoline was drawn from the disabled motors of the wreck and added to the supply. “It seems lonesome, don’t you know,” the Englishman said, as he took his seat on the Ann, “to go away and leave that poor fellow all alone in the valley, with no companionship save that of the stars and the wind!” “It gives me a shiver to think of it!” declared Ben. “Well,” Jimmie said in a tone far more serious than was usual with the boy, “every step he has taken since his birth has tended to this place. A million years ago, it was decreed that he should lie here, and that’s all there is of it!” “Quite true, quite true!” agreed the Englishman. “Aw, you can’t make me believe a man’s life is mapped out for him like that!” declared Carl. “I guess a fellow has some show!” When the boys reached the camp the eastern sky was ruddy with the approach of sunrise, and Mr. Havens sat well wrapped in blankets before the fire. His face was pale and showed suffering. “I thought you’d never come back!” he said. “I saw one of the machines drop, but I couldn’t for the life of me tell which one it was.” “Two of them dropped,” Ben explained, and in a short time the story of the adventures of the night was told. “It seems wonderful,” Mr. Havens said, “that we should drop into a region, almost by accident, whither so many things connected with the Kuro case were tending. When the Englishman brought the bag, I thought that the most remarkable occurrence in the world. But now the man who stood in the corridor at Colleton’s door seems to
- 36. lie over yonder in the valley. It seems like a chapter out of a fairy book!” “Why, it’s all simple enough!” Jimmie argued. “In fact, it’s the most commonplace thing in the world. This big man stripped Colleton of his disguise in the stateroom and put the articles into the bag, intending to throw it off the train the first time he got a chance. He set the bag out into the corridor or the vestibule so it would be handy when the right time came and the porter stole it.” “Is this a new edition of the dream-book?” asked Carl. “Then DuBois lost his hand-bag, and asked the porter to provide him one. For all we know the man just killed may have stolen the Englishman’s bag for his own use. Anyway the porter brought DuBois the bag he stole from the man who has just been killed.” “Go on!” advised Ben with a grin. “The porter neglected to remove the contents of the bag, and so the articles used in the disguise of Colleton come into the possession of the purchaser. The Englishman sets out on a hunting trip in the Rocky mountains, strays away from his companions, and turns up at the smugglers’ place with the bag in his hands.” “You’re only relating the obvious now,” Ben criticised. “And then,” Jimmie went on, “the big man brings Colleton into some hiding-place in the mountains, using an aeroplane as a means of communication with the cities. His machine is spied by boys who think their own machines can go some and the race follows. The big man drops his aeroplane into a hole in the air and is killed. The Englishman who bought the stolen bag, recognizes him as the man in charge of the sick man in the stateroom. Now, if that isn’t all perfectly simple, I don’t know what is!” “You take it for granted that Colleton is hidden in this vicinity, then?” asked Ben. “If he wasn’t, the big man wouldn’t have shown up here!” “When the big man came in and landed his aeroplane on the other side of the ridge,” Ben suggested, “he brought two men with him. When we went up in the Louise we saw two men walking about the ledge with lanterns in their hands.” “One of them may be Colleton!” shouted Carl.
- 37. “I don’t know about that,” Jimmie went on, “but I’ll tell you there’s some connection between the bunch that stole Colleton and the bunch the Canadian officers arrested for smuggling whiskey over the Canadian border. I don’t believe the red and green signals we saw night before last were entirely for the benefit of the smugglers. I’ll bet the big man who was killed because he didn’t know how to bring a machine out of an air-hole knew the language of those red and green lights!” Mr. Havens was assisted back to his tent, and the boys busied themselves getting breakfast. The Englishman wandered about the camp for a long time without speaking. It seemed to the boys that he was studying over the events of the night. Jimmie even suggested to Carl that the Englishman might be searching his memory for some incident connected with the journey across the continent which would place him in the possession of additional information concerning the man who had been killed. When breakfast was ready, the Englishman took his seat by the white cloth spread on the grass but ate sparingly. “Have you lost your appetite?” asked Carl. “That was quite a shock, don’t you know!” was the answer. “Are you sure the man we buried is the man who occupied the stateroom on the Pullman-car with the sick man?” asked Ben. “Quite sure!” was the slow reply. “Did you notice him talking with any one in the car?” asked Jimmie. “Indeed he was quite intimate with one of the travelers,” the Englishman replied. “They went to the smoking room together and played cards frequently. They were quite chummy, don’t you know.” “Would you know this second man if you saw him again?” “Why, of course,” answered the Englishman. “This second man, Neil Howell, is the gentleman who formed the hunting party I joined at San Francisco. He was quite anxious for me to go with him, don’t you know.” “When did you leave your party?” asked Ben. “Early yesterday morning,” was the reply. “I wandered about in the mountains until I came to the camp-fire where I was found.”
- 38. “Could you make your way to your camp now?” asked Jimmie. The Englishman shook his head. “It is in some of the wrinkles of the mountains,” he said, “but I couldn’t even make up my mind which way to set out if I started to find it.” “Your sense of direction must be deficient!” suggested Carl. “It must be!” was the answer. “You see,” he went on, “I wandered around this way and that, so long that I couldn’t tell whether my camp was east, west, north or south. During the last few hours of my wandering I was half dazed with hunger and fatigue, so there is little hope of my being able to locate the camp of my friends.” “Well, we can find it all right!” Jimmie declared. “I can take you up in the machine after we get done breakfast, and after we get last night’s kinks out of our systems, and we can find your camp if it’s anywhere within a thousand miles.” The Englishman appeared thoughtful for some moments before making any reply. Jimmie nudged Carl and whispered: “Look here, Cully, I don’t believe he wants to find that camp again! I don’t believe he wants to go back!” “Yes,” returned Carl, “the quiet, peaceful, uneventful life we are leading seems to appeal to him!” “We may be able to find the camp,” the Englishman said after a pause, “but really, you know,” he went on, “I wouldn’t want to take another ride in the air to-day!” “Oh, we can go to-morrow just as well,” laughed Jimmie. After breakfast the boys advised the Englishman to spend most of the day in sleep. They had had another hard night, and were in need of rest themselves. It was a warm, sunny day, and the lads, well wrapped in blankets, slept until almost noon. After they awoke and prepared dinner, Mr. Havens noticed Carl and Jimmie looking longingly in the direction of the machines. “What’s on now, boys?” he asked. “I want to find the answers to two questions,” Jimmie replied. “Where are the answers?” asked the aviator. “In the air,” grinned the boy. “What are the questions?” continued Mr. Havens.
- 39. “The first one is this: Who are the men the dead man brought in with him last night?” “And the other one?” “Where is the Englishman’s camp?” “Two very pertinent questions!” suggested Mr. Havens. “There’s another question,” Jimmie continued, “that I want the answer to, but I don’t see how I’m going to get it right away.” “Perhaps I can answer it!” “I’ll give you a try at it,” Jimmie laughed. “Well, what is it?” “Did the Englishman accidentally lose his camp or did he lose it on purpose? Can you answer that question?” “I’ve been watching the Englishman for some time,” the aviator replied, “and I think I can give you the answer. He left it on purpose!” “I noticed,” Jimmie said, “that he didn’t seem very anxious about my helping him find it!” “Well, whether he wants to find it or not,” Mr. Havens continued, “I must insist on you boys locating it!” “You want to know about this man Neil Howell!” laughed Jimmie. “Perhaps you have a notion that by finding him we can get track of the dead man’s associates. You want to know why he induced DuBois to make the mountain trip. In fact, there’s a whole lot of things you want to know about Neil Howell.” “That’s just the idea,” Mr. Havens replied. “I’m certain that DuBois left the camp voluntarily. There might have been a quarrel, for all I know. I half believe, also,” he continued, “that the Englishman knew what the bag contained when he left camp with it.” “I don’t know about that,” Jimmie replied, “but I do know that a man going out for a walk in the mountains wouldn’t be apt to carry a hand-bag with him if he intended to return.”
- 40. CHAPTER XVI. STILL ANOTHER GUEST. “You bet he wouldn’t!” declared Carl, who had come into the tent during the progress of the conversation. “He’d be more apt to carry a gun! What did he want to lug his toilet articles away for?” “Perhaps he wanted to get that bag out of camp!” suggested Jimmie. “What’s the answer to that?” asked Carl. “Suppose this Neil Howell recognized that bag as one formerly owned by the man he played cards with?” “That’s another dream!” Carl laughed. “Anyhow,” Jimmie said, “I’m going up in the Louise and find that camp!” “And I’m going with you,” Carl grinned. “Can’t I go anywhere without one of you boys tagging along?” demanded Jimmie in mock anger. “It’s a shame for you to say such things!” declared Carl. “After the number of times we’ve saved your life!” “All right!” laughed Jimmie. “Come along if you want to!” “If I were you,” Mr. Havens advised, “I wouldn’t try to land near the camp if you succeed in locating it. The song of the motors can be heard a long way off, you know, and the campers will be sure to know that an aeroplane is in the vicinity.” “That’s a good idea!” Carl agreed. “We ought to find the camp and sail over it, and around it, and then duck away as if we belong out on the Pacific coast somewhere. Then we can go back on foot, if it
- 41. isn’t too far away, and see what sort of a crowd the Englishman traveled with.” “That’s my idea of the situation,” Mr. Havens said. “And we ought not to say anything to the Englishman about where we’re going!” Jimmie suggested. “Because he’ll be eager to know what we find out, and may decide not to remain with us at all after we discover why he left his companions.” “We don’t know that he hasn’t told the absolute truth about his departure from camp,” Mr. Havens suggested, “but it will do no harm to work on the theory that a man merely in quest of mountain adventure would not leave his camp carrying a hand-bag. As Carl says, he’d be more likely to carry a gun!” Ben came into the tent and stood listening to the conversation. He agreed with the others that there was something queer about the Englishman’s sudden appearance with the hand-bag, but said that the fellow had really possessed a gun when he reached the fire where he had been found. “He told me,” Ben went on, “that Crooked Terry had taken his gun and other articles, including his money, from his person.” “Why didn’t you snatch Crooked Terry bald-headed and make him give ’em up?” asked Jimmie. “Because DuBois didn’t tell me about his being robbed until after we had left the crook asleep in the cavern. I think, by the way,” Ben continued, “that I’d better go up to the smugglers’ den to-day and see what I can learn regarding those two men.” “Is this a conspiracy to leave me all alone in the camp again?” asked Mr. Havens. “I’m getting about enough of solitude.” “Why, there’s the Englishman,” suggested Jimmie. “Don’t you ever think he won’t want to go, too,” Ben laughed. “He’s the craziest man about flying machines I ever saw.” “But early this morning,” Jimmie argued, “he said that he didn’t care about going into the sky again to-day.” “Perhaps that’s because you suggested hunting up his camp,” laughed Ben. “Somehow he don’t seem to want to find that camp.” “Suppose,” suggested Mr. Havens, “you boys go in relays. Let Jimmie and Carl go and look up the camp first, and after they return
- 42. Ben and DuBois can visit the smugglers’ camp.” “That’s all right,” Ben exclaimed. “I’ll remain here until Jimmie and Carl return, if they’re not gone too long!” “Did you see anything of intruders while we were gone?” asked Jimmie turning to Mr. Havens. “Why,” replied the aviator, “I did see a man looking toward the camp from the valley to the north, but no attempt to molest me was made.” “So that’s why you don’t want to be left alone!” laughed Jimmie. “You think perhaps those fellows are hanging around here yet!” “They may be, at that!” Carl suggested. “We have the faculty of getting into a storm center,” Jimmie complained. “We get a collection of humanity around every camp we make! If we should go and make a camp on top of the Woolworth building, in little old New York, people would be making a hop-skip- and-jump from the sidewalk and inviting themselves to dinner!” “Well, go on out and stir up another mess of visitors,” laughed Mr. Havens. “And when you find this camp,” he added, “don’t land anywhere near it and try to creep in on the campers. All you’ve got to do is to come back and tell us where it is!” “All right!” laughed Jimmie. “I’ll make a map of the country so any one can find it.” The two boys were soon away in the Louise, and then Ben and the Englishman went to Mr. Havens’ tent to further talk over the situation. The millionaire was very much inclined to ask the Englishman just why he had left his camp, but finally decided not to do so. DuBois was very thoughtful and not inclined to join in the conversation. More than once they saw him step to the flap of the tent and look out over the valley. On such occasions he seemed nervous and anxious. “Are you expecting company?” Ben asked after one of these visits. “I heard some talk about people watching the camp, don’t you know,” the Englishman replied, “and it rawther got on me mind!” “There won’t any one come here in the daytime,” Ben urged.
- 43. “Did you see the faces of the men who came this morning?” asked the boy turning to Mr. Havens. “I didn’t say that I saw men,” smiled the aviator. “I said that I thought I saw a man looking toward the camp.” “Did you see his face?” insisted the Englishman. “I did not!” was the reply. “Can you describe him in any way?” “I’m afraid not!” The Englishman walked to the flap of the tent again and looked out. “For instance,” he said looking back into the tent, “was the general appearance of the fellow anything like the general appearance of the man who is approaching the fire from the other side?” The aviator gave a quick start of surprise and Ben sprang to his feet and walked out to the fire, closely followed by the Englishman. The man approaching from the south was evidently not a mountaineer. He was remarkably well-dressed, although his garments showed contact with mountain thickets, and his walk was unsteady and like that of one unfamiliar with rough ground. He wore a derby hat, a silk tie, and a gold watch-chain traversed his vest from left to right. He was, in fact, about the cut of a man one would expect to meet in the business district of New York. Instead of watching the visitor, Ben turned his eyes toward the Englishman, determined to see if any signs of recognition showed on the face of the latter. His first impression was that this man had in some way found his way there from the camp which the Englishman had deserted. DuBois’ face expressed only curiosity and surprise as the visitor came closer to the fire. Ben turned to the newcomer. “Good-afternoon!” he said. “Same to you!” replied the other. “You can’t understand,” he added with a faint smile, “how glad I am to see once more a face that reminds me of civilization.” “That’s me!” laughed Ben winking at the Englishman. “That’s both of you, and the man in the tent, too!” laughed the other. “I’ve been wandering around this everlasting, eternal,
- 44. Providence-forsaken valley for three or four days, living on ground squirrels and seeking to become intoxicated on river water.” “Did you lose your camp, too?” asked Ben with a chuckle. “I never had any camp in this country!” was the reply. “I came in by way of Crow’s Nest, with a pack of provisions on my back, looking for land worth squatting on. I ate my provisions the first week, lost my way the second, and traveled on my nerve the third.” “Did it make good going?” asked Ben with a grin. “Fairly good!” was the reply. “You see,” he went on, “I had a couple of automatic guns and plenty of cartridges, so I’d shoot red ground squirrels when ever I got hungry and build a fire in among the tall trees and cook ’em. Then I’d go to sleep by the fire and wake up that night, or the next morning, or the day after the next morning, or any old time. And that’s the kind of an existence I’ve been having.” “That’s the wild, free life, all right!” Ben agreed. “I’ve been chased by bears, and kept awake at night by lynxes, and wolverines, until it seems to me as if I had butted into the Central Park Zoo! And right this minute,” he added, looking around the camp with wistful eyes, “I’m about as hungry as a human being can be and stand on his feet. I haven’t had a drop of coffee for a month!” “I was waiting for that!” Ben grinned as he moved toward the coffee-pot and provision box. “Everybody that comes here is hungry! I’ve got so I make a break for the coffee-pot and the grub the minute I see a stranger approaching.” “I’m glad you’ve got the habit,” laughed the other. “I’ve butted into camps in this country before now where a man wasn’t welcome to take a second breath out of the atmosphere!” “Recently?” asked Ben. “Why, only three or four days ago,” the stranger answered, “I struck a camp where they had tons and tons of provisions, and they wouldn’t give me the second meal! Yes, sir, they fired me out after I’d had a few egg sandwiches and a cup of coffee substitute.” “How long ago was this?” asked Ben, glancing quietly at the Englishman.
- 45. “Three or four days ago!” was the answer. “I’ve been traveling nights to keep warm, and to keep out of the clutches of the wild animals, and sleeping days so long that I’ve lost all track of time. It may have been three days ago and it may have been four days ago.” “Can you give me the direction of this camp?” asked the Englishman. “I’d like to know something about the fellows there, if you don’t mind.” “Oh, I don’t know which way it is from here. I couldn’t find it if I wanted to, and I’ll give you a straight tip right now that I don’t want to! Just for company’s sake, understand, I tried to get a night’s sleep within sight of their camp-fire. I rolled myself in a blanket and was just dreaming that I was eating a porterhouse steak at Sherry’s, when the midnight concert at the camp began. I guess they were all good and drunk before morning.” “Do you know,” began the Englishman, “that I half believe that you found the camp I belonged in!” “If you were in the camp when I tried to sleep near it,” the stranger went on, “you probably got a good souse before morning.” The Englishman turned away to the tent, and Ben busied himself in preparing dinner for the stranger who gave his name as Martin Sprague. “I see,” Sprague went on, while the dinner cooked, “that you boys have a couple of fine flying machines. Was that your machine that lit out over the valley a short time ago? When I saw that machine, I said there must be a camp in this side of the valley, so I followed my nose and here I am.” After a time, Ben placed a substantial meal before Sprague and then, to an answer to a gesture from the Englishman, hastened back to the tent. “Do you know,” DuBois said, as the two stood together at the flap, “that fellow who just came in was with Neil Howell in San Francisco! I saw the two together there often. If he went to our camp, he found Neil Howell there, and he received no such treatment as he reports.” “Then you think the fellow’s a fraud, do you?” asked Ben.
- 46. “I don’t know about that!” the Englishman replied, “but I do know that he is trying to deceive you, and my private opinion is that he came to this camp for a purpose, and with the consent of Neil Howell.”
- 47. CHAPTER XVII. CARL GETS INTO TROUBLE. The sun shone warm on the planes of the Louise as Jimmie and Carl sailed over the broken country to the west of the camp. They passed a ridge so high that the timber line broke a couple of hundred feet below the summit, and then dropped, shivering, into a depression wider but not so green as the one in which their tents stood. The boys were taking their time, and, in the low altitude of the valley, conversation was possible as they moved along, looking to right and left for some sign of a camp. “The Englishman’s friends ought not to be much farther away,” suggested Carl, after an hour. “We are at least fifteen miles from our tents already.” “Yes,” agreed Jimmie, “the ridge we crossed takes up a good deal of room. If they are not in this wrinkle, they may be in the next one.” “Wrinkle is exactly the word,” Carl grinned. “This country looks as if some one had taken a level plain and crowded it together until the surface broke into seams and crags. It makes me think of the undulating surface of an old boot!” The boys traversed the valley from north to south but saw no indications of tents or camp-fires. The ridge to the west ran out at the north end of the valley, and the boys turned there, preferring not to ascend into the cold air again unless it became necessary. The valley in which they now found themselves ran in a northeasterly direction and broke into a canyon at the end farthest
- 48. to the east and north. The boys turned as they swung around the point of rock and whirled along the new depression. Presently Carl caught his chum by the arm and handed him the field-glass with which he had been looking over the country. Jimmie used the glass for a moment and then turned back to Carl with a pleased look on his freckled face. “You know what that is, don’t you?” he asked. “Sure!” Carl answered. “That’s the north end of our own valley, we see,” Jimmie went on, “and the shelf we have just come in sight of is the one from which the red and green signals were shown night before last.” “That’s right!” grinned Carl. “Then, don’t you see,” Jimmie went on, “the signals were made for the benefit of some one in this valley.” “That’s the idea!” Carl chuckled. “Now, suppose we find the tent the Englishman left in this vicinity,” the boy went on, “what would that mean?” “It would suggest to me,” Carl replied, “that the signals were made for the benefit of some one in that camp.” “Right-o!” replied Jimmie. “But where is this blooming camp?” Carl asked. “We’ll find it here somewhere!” Jimmie answered, confidently. Directly the boys came to a canyon which opened at the west of the valley and led to a grassy plateau higher up. At some distant time the place now occupied by the plateau had doubtless been an enlargement and extension of the canyon. However, as the years passed, the rocks had crumbled under the action of water until the great dent had become filled. One look to the left as the boys moved slowly past the mouth of the canyon was sufficient. A fire was blazing high in the center of the plateau and half a dozen tents were scattered about. On every side the walls of rock came down to the green grass which lay like a carpet over the floor of the plateau. Here and there the boys saw dark openings in the walls, similar to the one they had observed at the smugglers’ camp.
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