Stratified Polyhedra
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252 David A. Stone Massachusetts Institute of Technology, Cambridge, MNUSA and State University of New York at Stony Brook, Stony Brook, NY/USA
Stratified Polyhedra
Springer-Verlag Berlin . Heidelberg . NewYork 1972
AMS Subject Classifications (1970): Primary: 57B05, 57C40, 57C50 Secondary: 55F65, 57F20
ISBN 3-540-05726-9 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05726-9 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin· Heidelberg 1972. Library of Congress Catalog Card Number 77-187427. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Introduction The present paper is a revised version of a much longer work (which had fewer results) entitled "Block Bundle Sheaves". This is an obsolete phrase for "stratified polyhedra"; and since my work has been referred to under the old title, I mention it as a sort of subtitle of the present work.
During the process of
revision, I profited greatly from lecturing in a graduate seminar at M.I.T., since it was necessary to describe technical definitions and procedures intuitively.
I talked about my difficulties to
many friends, especially Ralph Reid, my sister Ellen and Dennis Sullivan.
Above all, I am indebted to Colin Rourke, who suggested
the present method of defining stratifications, and who urged me to change terminology to conform better with Thorn's theory [28] of "ensembles stratifies".
I am also glad to express my gratitude
to William Browder. The two facts about block bundles, as defined by Rourke and Sanderson, which make them so important are:
t , submanifold of Q, then M has a normal block bundle in
1.
if M is a p.
Q,
which is in some sense unique;
2.
there is a relative transversality theorem: if M.
Q.
then there is an arbitrarily small isotopy f
t
block transverse to M; bdy M in bdy Q,
N are subrnanifol.ds of
of Q such that f N is 1
moreover if bdy N is already block transverse to
then we may require f to be the identity on bdy Q. t
That M has a block bundle neighbourhood in Q was shown by a simple construction: roughly speaking, take a simplicial triangulation B of Q in
IV which M is covered by a full subcomplex A. subdivision of B.
For each simplex SEA,
* tk(s. B') s*A = b * l.k(s. At) s*B =
let
(the simplicial join);
'"
where s E B' is the barycentre of s ,
f s*B
Let B' be a first derived
Then s*B is a block over s*A.
and
: s E AJ forms the required block bundle. This construction can equally well be performed when M and Q are
no longer manifolds but general polyhedra.
It seemed reasonably to h
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