Thickenings
From Manifold Atlas
Revision as of 13:41, 27 November 2010 by Diarmuid Crowley (Talk | contribs)
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
Tex syntax errorwith boundary such that:
- the map is an isomorphism,
-
Tex syntax error
is homotopy equivalent to .
Tex syntax errorthickens . If there is such a manifold
Tex syntax error, we would like to know how many up to homeomorphism or diffeomorphism if
Tex syntax erroris smooth.
In [Wall1966a] Wall introduced the notion of a thickening, defined below, to investigate the questions raised above. This page summarizes the basis results concerning thickenings.
Recall that or denotes respectively the topological, piecewise linear or smooth categories.
Definition 0.1 [Wall1966a, Section 1]. Let be a finite connected CW complex. An -dimensional -thickening of consists of
- a compact -dimensional -manifold
Tex syntax error
with connected boundary such that - a basepoint and an orientation of ,
- a simple homotopy equivalence .
Two thickenings and are isomorphic if there is a -isomorphism preserving and the orientation of and such that is simple homotopic to . In particular there is a simple homotopy commutative diagram
Tex syntax error
2 References
- [Wall1966a] C. T. C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR0192509 (33 #734) Zbl 0149.20501