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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)



Strongly stratified homotopy theory

Author: David A. Miller
Journal: Trans. Amer. Math. Soc. 365 (2013), 4933-4962
MSC (2010): Primary 54E20; Secondary 55R65
Published electronically: March 4, 2013
MathSciNet review: 3066775
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Abstract: This paper concerns homotopically stratified spaces. These were defined by Frank Quinn in his paper Homotopically Stratified Sets. His definition of stratified space is very general and relates strata by ``homotopy rather than geometric conditions''. This makes homotopically stratified spaces the ideal class of stratified spaces on which to define and study stratified homotopy theory.

We will define stratified analogues of the usual definitions of maps, homotopies and homotopy equivalences. Then we will provide an elementary criterion for deciding when a strongly stratified map is a stratified homotopy equivalence. This criterion states that a strongly stratified map is a stratified homotopy equivalence if and only if the induced maps on strata and holink spaces are homotopy equivalences. Using this criterion we will prove that any homotopically stratified space is stratified homotopy equivalent to a homotopically stratified space where neighborhoods of strata are mapping cylinders. Finally, we will develop categorical descriptions of the class of homotopically stratified spaces up to stratified homotopy.

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David A. Miller

Keywords: Stratified space, homotopy link
Received by editor(s): April 24, 2011
Received by editor(s) in revised form: December 23, 2011
Published electronically: March 4, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.