Table Of ContentLecture Notes ni
Mathematics
detidE yb .A Doid dna .B Eckmann
834
IIIIIIIIIIIIIIII IIIIIIIIIII IIIIIIIIIIIII I IIIIIIIIII I IIIIIII IIIIII
Geometric ygolopoT
Proceedings of the Geometric Topology Conference
held ta Park City, Utah, February 19-22, 1974
Edited yb .L C. Glaser dna .T .B gnihsuR
ill,, IIIIIIII IIIIIII II IIII IIII
galreV-regnirpS
Berlin. Heidelberg- New York 1975
Prof. Dr. Leslie Curtis Glaser
Prof. Dr. Thomas Benjamin Rushing
Department of Mathematics
The University of Utah
Salt Lake City, UT 84112/USA
Library of Congress Cataloging in Publication Data
Geometric Topology Conference~ Park City, Ut&h, 1974.
Geometric Topolog~r Conference~ Park City, Utah~
February 19-22~ 1974.
(Lecture notes in mathematics ; 438 )
Bibliography: p.
Includes index.
i. Topology--Congresses. I. Glas~er, Leslie C.,
1937- ed. II. Rushing, T. Benny~ ed. III. Se-
ries: Lecture notes in mathematics (Berlin) ; 438.
QA3.L28 no. 438 [QA611L.AZJ 510'.8s [514'.3]
74-34326
AMS Subject Classifications (1970): 22-oz 22co5, 54.o2, 54c15, 54c25,
54C55, 54C60, 54D05, 54F45, 54H15,
~5-02, 52A25, 25802, 55c10, 52c25,
55D10, 55D15, 55D35, 55D99, 55F05,
55F10, 57~02, 57AXX, 57C05, 57C30,
57C35, 57C45, 57C52, 57E05, 58-02,
58 D05, B05, 58 58 H05.
tSBN 3-540-07137-7 Springer-Verlag Berlin • Heidelberg • New York
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PREFACE
These are the proceedings of the Geometric Topology Conference sponsored
jointly by the National Science Poundation and the University of Utah. The
conference was held at Park City, Utah which is an old mining town and recrea-
tion area located about twenty-five miles from Salt Lake City. The consulting
committee for the conference consisted of C.E. Burgess, University of Utah,
R.J. Daverman, University of Tennessee, R.D. Edwards, U.C.L.A., L.C. Glaser,
University of Utah, and R.C. Kirby, Berkeley, with T.B. Rushing, University of
Utah, serving as organizer.
The conference was focused on various aspects of geometric topology, including
topological topology, piecewise linear topology, differential topology, infinite
dimensional topology and 3-space topology. The present volume consists mainly of
papers presented at the conference, however a few related papers are also
included.
Although great strides have been made in geometric topology over the past
few years, it was apparent at the conference that the area is still extremely
active and will remain so for many more years. Many questions of current
interest came to light at the conference. Most of those questions are either
explicit or implicit in the articles of this proceedings.
This entire volume was typed by Ms. Kathi Green on a IBM magnetic card
selectric typewriter. For her remarkable patience and dedication, the editors
wish to express their sincere appreciation. All of the illustrations as well as
the final editing and proofreading was done by L.C. Glaser. This editor, in
particular, is deeply indebted to Ms. Green. If she had not been as competent
and one of the best typists he has ever had the pleasure of working with, he
could have never done it alone otherwise.
L.C. Glaser
T.B. Rushing
Editors, October, 1974
TSIL FO STNAPICITRAP
A( * after a name indicates an invited speaker)
Selman Akbulut M.L. Curtis*
.U of Cal. at Berkeley Rice University
Rick Ancel Bob Daverman*
University of Wisconsin University of Tennessee
Steve Armentrout* P.H. Doyle
Penn State University Michigan State University
Jim Bailey Paul Duvall
U. of B.C. Oklahoma State University
Victor Belfi Bill Eaton*
TCU University of Texas
Fred Benson Bob Edwards*
University of Utah UCLA
Harry Berkowitz Benny Evans
Sun Shipbuilding .oC Oklahoma State University
R.H. Bing* Dan Everett
University ofT exas University of Wisconsin
Lois Broussard Lawrence Fearnley
University of Texas UYB
Beverly Brechner Steve Ferry
University of Florida University of Kentucky
Morton Brown Ben Freed
University of Michigan Kent State University
John Bryant* Mike Freedman*
Florida State University University of California at
Berkeley
C.E. Burgess
University of Utah Chuck Feustel
VPI
J.W. Cannon*
University of Wisconsin Ross Geoghegan
SUNY-Binghamton
Jim Cantrell*
University of Georgia Richard Gillette
Montana State University
Eliot Chamberlin
University of Utah Dave Gillman
UCLA
T.A. Chapman*
University of Kentucky Les Glaser
University of Utah
Don Coram
Oklahoma State University Herman Gluck*
University of Pennsylvania
IV
Deborah Goldsmith Duane Loveland
University of Chicago Utah State University
Mike Handel Erhard Luft
U. of Cal. at Berkeley U. of B. C.
Bill Hayer Louie Mahoney
University of Tennessee SUNY-Binghamton
J.P. Hempe I* Joe Martin*
Rice University University of Wyoming
John Hocking Jim Maxwell
Michigan State University Oklahoma State University
L.S. Hush* Jim Mayland, Jr.
University of Tennessee York University
Ed Hutchings Tom McAlone
U. of .B .C .U of Cal. at Berkeley
Marvin Israel D.R. McMillan, Jr.*
U. of 111. Urbana University of Wisconsin
W.H. Jaco* Tom McMillan
Rice University University of Utah
Steve Kaplan Paul Melvin
.U of Cal. at Berkeley .U of Cal. at Berkeley
Louis Kauffman Mark Meyerson
U. Ill. at Chicago Cr. Stanford University
J.E. Keesling* Ken Millett
University of Florida .U .C at Santa Barbara
R.C. Kirby* Jack Neuzil
U. of Cal. at Berkeley Kent State University
R.C. Lacher* Victor Nicholson
Florida State University Kent State University
Howard Lambert David Nordstrom
University of Iowa U. of Cal. at Berkeley
Jack Lamoreaux Richard Osborne
UYB Colorado State University
Laura Leach Erik Pedersen
University of Wyoming University of Chicago
V.T. Liem Carl Pixley
University of Utah University of Texas
Lloyd Lininger Tom Price*
SUNY-Albany University of Iowa
IIV
Frank Quinn* Bruce Treyhig
Yale University Texas A and M University
Gerhard Ritter Edward Turner
University of Florida SUNY-Albany
Dale Rolfsen Ronald Urner
U. of .B .C U. of Wyoming
Dennis Roseman Jim Van Buskirk
University of Iowa University of Oregon
Harry Row Gerald Venema
University of Tennessee University of Utah
Leonard Rubin Bill Voxman
University of Oklahoma University of Idaho
Benny Rushing Bruce Welmers
University of Utah ALCU
Martin Scharlemann Alden Wright
.U of Cal. at Berkeley Western Mich. U.
Dick Schori
LSU
Arlo Schurle
.U of N. .C at Charlotte
Peter Scott
Minnesota and Liverpool .U
Jack Segal
University of Washington
C.L. Seebeck, IIl*
Michigan State University
Julius .L Shaneson*
Rutgers University
Neelima Shirikhande
University of Wyoming
Mike Siemon
.U of Cal. at Berkeley
Brian Smith
.U of Southwestern .aL
Joseph Taylor
University of Utah
Jeff Tollefson
Michigan State University
CONTENTS
Armentrout, Steve: Decompositions and Absolute Neighborhood Retracts ..... 1
Benson, Fred: A Short Proof of a Kirby Flattening Theorem .......... 6
Berkowitz, H.W. and Roy, Prabir: General Position and Algebraic
Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Bing, R.H,: Vertical General Position . . . . . . . . . . . . . . . . . . . . 16
Burgess, C.E.: Semicellular Sets in Crumpled Cubes ............. 42
Bryant, J.L., Lacher, R.C., & Smith, B.J.: Free Spheres With Mapping
Cylinder Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . 58
Cannon, J.W.: Taming Cell-like Embedding Relations .............. 66
Cantrell, J.C.: Some Non-continuous Functions and Pseudo-flattenings ..... 119
Chapman, T.A.: Homotopic Homeomorphisms of Hilbert Cube Manifolds ...... 122
Curtis, Morton: A Homotopy Theoretic Approach to Lie Groups ......... 137
Daverman, Robert J.: On the Absence of Tame Disks in Certain Wild Cells . . . 142
baverman, Robert J.: A Summary of Results and Problems Concerning
Flatness of Codimension One Spheres in n E ............... 156
Eaton, William and Pixley, Carl: I S Cross a UV ~ Decomposition of
S 3 Yiel• ds S I x S 3 166
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Edwards, Robert: Demension Theory, I . . . . . . . . . . . . . . . . . . . . . 19S
Freedman, Michael H.: Automorphisms of Circle Bundles Over Surfaces ..... 212
Glaser, Leslie C.: On Mildly Cellular Pseudo Cells .............. 21S
Gluck, Herman: Almost All Simply Connected Closed Surfaces Are Rigid ..... 225
Guti6rrez, M.A. and Lacher, R.C.: Semifree Group Actions and Homology
Spheres 240
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Handel, Michael: Approximating Stratum Preserving CE Maps Between
CS Sets by Stratum Preserving Homeomorphisms .............. 245
Hempel, John: One Sided Incompressible Surfaces in 3-manifolds ........ 251
Hermion, Geoffrey and McPherson, J.: A Local Factorization Theorem
For Arcs With Finite Penetration Index ................. 259
Husch, L.S,: Regular zn-actions . . . . . . . . . . . . . . . . . . . . . . . 276
Jaco, William: Roots, Relations and Centralizers in Three-manifold
Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Kirby, R.C. and Siebenmann, L.C.: Normal Bundles for Codimension 2
Locally Flat Imbeddings . . . . . . . . . . . . . . . . . . . . . . . . . 310
Keesling, James: The Cech Homology of Compact Connected Abelian
Topological Groups With Applications to Shape Theory .......... 325
Lacher, R.C.: k-sphere Mappings on S 2k+l .................. 332
Lininger, Lloyd: Codimension 1Foliations on Manifolds With Even Index .... 336
Mayland, Jr., E.J.: The Residual Finiteness of the Groups of Classical
Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933
McHillan, Jr., D.R.: Cutting Off Homotopies on Acyclic Sets . . . . . . . . . 343
Millett, Kenneth C.: Homotopy Groups of Automorphism Spaces ......... 3S3
Price, T.M.: On Codimension Two Embeddings .................. 365
Price, T.M. and Seebeck III, C.L.: A Codimension Two Taming Theorem ..... 371
Quinn, Frank: Semifree Group Actions and Surgery on PL Homology Manifolds . . 395
Rolfsen, Dale: A Surgical View of Alexander's Polynomial ........... 415
Rushing, T.B.: A ~ummation: the Compacta X in n S ~or Which
Sh~) : Sh(S ) is equivalent to n S -X n ~ - S S ........... 424
Seebeck III, C.L.: Locally Homotopically Unknotted Embeddings of
Manifolds in Codimension Two . . . . . . . . . . . . . . . . . . . . . . 427
Shaneson, Julius L.: Spines and Spinelessness ................ 431
Wright, Alden H.: Monotone Mappings and Degree One Mappings Between
PL Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
DECOMPOSITIONS AND ABSOLUTE
NEIGHBORHOOD RETRACTS
BY
STEVE ARMENTROUT
,1,NOITCUDORTNI
Upper semicontinuous decompositions have been used to con-
struct a number of retracts.
In this paper, we shall be interested in studying this process with particular
interest in applications to the following well-known problem [5, p. 218] in the
theory of retracts:
,MELBORP
Does each absolute neighborhood retract have the homotopy type of a
polyhedron? It is known ,5[ .p 106] that each absolute neighborhood retract is
homotopically dominated by a polyhedron. De Lyra ]6[ has shown that each simply
connected absolute neighborhood retract has the homotopy type of a polyhedron.
We shall consider only compact metric retracts, and we shall use the notations
AR and ANR for (compact metric) absolute retracts and absolute neighborhood re-
tracts, respectively. By a POLYHEDRON we shall mean a compact polyhedron.
If X is a space and G is an upper semicontinuous decomposition of X, then
,X/G denotes the associated decomposition space, and @:X ~- X/G denotes the projec-
tion map.
,2 BASIC THEOREMS,
,MEROEHT
(Smale [8], Borsuk ,S[ p. 131]). Suppose that X is an ANR, O
is an upper semicontinuous decomposition of X into absolute retracts, and X/G is
an ANR. If X is an AR, then so is X/G.
PROBLEM,
Does the result hold without the hypothesis that X/G be finite
dimensional?
A compact set is CELL-LIKE if and only if it has a cellular embedding in some
euclidean space.
THEOREM, (Lacher [7]). If X is a finite dimensional ANR, G is an upper
semicontinuous decomposition of X into cell-like sets, and X/G has finite dimen-
sion, then X/G is an ANR. If X is an AR, 8o is X/G. Further, X and X/G
have the same homotopy type.
If n is a positive integer, a compact set M in a space X has PROPERTY
UV n if and only if for each open neighborhood U of M, there is an open neighbor-
hood V of M such that if 0 <_ k ~ n and k + f V S : is a map, then f ~ o in
U. M has PROPERTY UV ~ if and only if for each open neighborhood U of M, there
is an open neighborhood V of M such that V ~ o in U. See, for example, [i].
EXAMPLES, .I Each AR in a finite dimensional ANR has property UV ~.
.2 Each cellular set in a manifold has property UV ~.
The following theorem is a corollary of results of [2].
THEOREM,
Suppose X is an ANR, G is an upper semioontinuou8 decomposition
of ,X and X/G has finite dimension .m Suppose each element of G has property
U~. Then X/G is an ANB. If X is an AR, so is X/G.
If X is an ANR, let A(X) be the minimum dimension of polyhedra that homo-
topically dominate X.
For the following result, see [2].
THEOREM. Suppose X is an ANR, G is an upper semicontinuous decomposition
of ,X and X/G is an ANR. Let k = max{ A )X( (X/G) A , .} Suppose that each element
of G has property U~. Then ~:X + X/G is a homotopy equivalence, and thus X
and X/G have the same homotopy type.
COROLLARY. If X is an ANR, G is an upper semicontinuous decomposition of
X into absolute retracts and X/G is finite dimensional, then X and X/G is
finite dimensional, then X and X/G have the same homotopy type.
,3 ANR's,
CONSTRUCTION OF If 2 B is a disc with boundary S ,I then Bz/S I
is an ANR; in fact, Ba/S ~ I S .2 We may relate this example to the theorems of
section 2 as follows: Let C(S )I be a cone over I S with a n B C(S )I = S .I Now
B2/S I ~ (B 2 u C(SI))/CCS I) ~ 2 S.
The ideas involved in the proof above yield a proof of a result on decomposi-
tions of ANR's where at most finitely many elements of the decomposition are ANR's
that are not AR's.
THEOREM i. Suppose X is an ANR and G is an upper semicontinuous decom-
position of X into points and finitely many ANR's ,IN 2 N ..... and N .m If X/G
is finite dimensional, it is an ANR.
