The moduli space of thickenings
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Abstract:
Fix $K$ a finite connected CW complex of dimension $\leq k$. An $n$-thickening of $K$ is a pair \[ (M,f) ,\] in which $M$ is a compact $n$-dimensional manifold and $f\colon K \to M$ is a simple homotopy equivalence. This concept was first introduced by C.T.C. Wall approximately 40 years ago. Most of the known results about thickenings are in a range of dimensions depending on $k$, $n$ and the connectivity of $K$.
In this paper we remove the connectivity hypothesis on $K$. We define moduli space of $n$-thickenings $T_n(K)$. We also define a suspension map $E\colon T_n(K)$ $\to T_{n{+}1}(K)$ and compute its homotopy fibers in a range depending only on $n$ and $k$. We will show that these homotopy fibers can be approximated by certain section spaces whose definition depends only on the choice of a certain stable vector bundle over $K$.
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Additional Information
- Mokhtar Aouina
- Affiliation: Department of Mathematics, Jackson State University, Jackson, Mississippi 39217
- Email: mokhtar.aouina@jsums.edu; mokhtarbenaouina@gmail.com
- Received by editor(s): March 26, 2010
- Received by editor(s) in revised form: May 16, 2011
- Published electronically: May 30, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 6689-6717
- MSC (2010): Primary 57R19; Secondary 55P10, 55P91, 55R25, 55Q99, 55S35, 55S40, 55U10, 55N99
- DOI: https://doi.org/10.1090/S0002-9947-2012-05645-5
- MathSciNet review: 2958952