Thickenings
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[edit] 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 basic results concerning thickenings.
Recall that or denotes respectively the topological, piecewise linear or smooth categories.
Definition 1.1 [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
[edit] 2 Constructions and examples
The simplest examples of thickenings come from -disc bundles with sections over manifolds, . Let be a closed -manifold of dimension , let be a bundle with fibre and with section . Then
- the pair is an -thickening of .
- the pair is an -thickening of a point.
[edit] 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. Clearly the isomorphism class of the bundle is an invariant of the thickening class . This if denotes the set of isomorphism classes of stable -bundles over we obtain a map
[edit] 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 4.1 [Wall1966a, Proposition 5.1] [Chazin1971, Theorem 1]. For all , the stable classifying map gives rise to a set bijection
[edit] 5 References
- [Chazin1971] R. L. Chazin, Stable thickenings in the topological category, Proc. Amer. Math. Soc. 29 (1971), 175–178. MR0296950 (45 #6009) Zbl 0214.22302
- [Wall1966a] C. T. C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR0192509 (33 #734) Zbl 0149.20501