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:
The set of isomorphism classes of -dimensional -thickenings over is denoted
2 Constructions and examples
The simplest examples of thickenings come from $q$-disc bundles with sections over manifolds, $q > 2$. Let $X$ be a closed $\Cat$-manifold of dimension $k$, let $M \to W$ be a bundle with fibre $D^q$ and with section $s \colon M \to X$.
- the pair $(M, s)$ is an $n$-thickening of $X$.
- the pair $(D^, pt)$ is an $n$-thickening of a point.
3 Invariants
An important invariant of a thickening is the induced stable -bundle over :
where is the stable -tangent bundle of . Given that stable bundles over the space are classified by maps to the classifying space one equivalently thinks of
where is the classifying map.
4 Classification
An extremely useful classification theorem in manifold theory is the classification of stable thickenings where originally due to Wall in the smooth catagory.
Theorem 3.1 \cite{} \cite{.} For all , the stable classifying map gives rise to a set bijection
5 References
- [Wall1966a] C. T. C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR0192509 (33 #734) Zbl 0149.20501