Thickenings
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1 Introduction
Let be a finite connected CW-complex of dimension
. For a given
we would like to know if there is a compact manifold
with boundary
such that:
- the map
is an isomorphism,
-
is homotopy equivalent to
.
In this case we say that thickens
. If there is such a manifold
, we would like to know how many up to homeomorphism or diffeomorphism if
is 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
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
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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