Thickenings
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# a basepoint $m \in \partial M \subset M$ and an orientation of $TM_m$, | # a basepoint $m \in \partial M \subset M$ and an orientation of $TM_m$, | ||
# a simple homotopy equivalence $\phi : K \to M$. | # a simple homotopy equivalence $\phi : K \to M$. | ||
− | Two thickenings $(M_0, \phi_0)$ and $(M_1, \phi_1)$ are ''isomorphic'' if there is a $\Cat$-isomorphism $f \colon M_0 \cong M_1$ preserving $m$ and the orientation of $TM_m$ and such that $f_0$ is simple homotopic to $\phi_1 \circ f$. | + | Two thickenings $(M_0, \phi_0)$ and $(M_1, \phi_1)$ are ''isomorphic'' if there is a $\Cat$-isomorphism $f \colon M_0 \cong M_1$ preserving $m$ and the orientation of $TM_m$ and such that $f_0$ is simple homotopic to $\phi_1 \circ f$. In particular there is a simple homotopy commutative diagram |
+ | $$ | ||
+ | \xymatrix{ | ||
+ | M_0 \ar[f]^{f} \ar[dr] & M_1 \ar[dl]\\ | ||
+ | K.} | ||
+ | $$ | ||
{{endthm}} | {{endthm}} | ||
Revision as of 13:41, 27 November 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
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