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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The moduli space of thickenings


Author: Mokhtar Aouina
Journal: Trans. Amer. Math. Soc. 364 (2012), 6689-6717
MSC (2010): Primary 57R19; Secondary 55P10, 55P91, 55R25, 55Q99, 55S35, 55S40, 55U10, 55N99
Published electronically: May 30, 2012
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Abstract: Fix $ K$ a finite connected CW complex of dimension $ \leq k$. An $ n$-thickening of $ K$ is a pair

$\displaystyle (M,f)\, ,$

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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05645-5
PII: S 0002-9947(2012)05645-5
Received by editor(s): March 26, 2010
Received by editor(s) in revised form: May 16, 2011
Published electronically: May 30, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.