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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
MathSciNet review: 2958952
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Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • [A-K] Aouina, M.; Klein, J.R.: On C.T.C. Wall's suspension theorem.
    Forum Mathematicum 18, 829-837 (2005).
  • [B] Becker, J.C.: Extension of cohomology theories.
    Illinois J. Math. 14, 551-584 (1970). MR 0273598 (42:8476)
  • [C-J] Crabb, M.; James, I.M.:
    Fibrewise homotopy theory. Springer Monographs in Mathematics. Springer-Verlag London, 1998. MR 1646248 (99k:55001)
  • [D-D-K] Dror, E.; Dwyer, W. G.; Kan, D. M.: Equivariant maps which are self homotopy equivalences.
    Proc. Amer. Math. Soc. 80, no. 4, 670-672 (1980). MR 587952 (82a:55009)
  • [G] Goodwillie, T. G.: Calculus. I. The first derivative of pseudoisotopy theory.
    $ \,K$-Theory 4, no. 1, 1-27 (1990). MR 1076523 (92m:57027)
  • [H] Hirsch, M.: Immersions of manifolds.
    Trans. Amer. Math. Soc. 93, 242-276 (1959). MR 0119214 (22:9980)
  • [Ho] Hodgson, J.P.E.: Obstructions to concordance for thickenings.
    Inventiones Math. 5, 292-316 (1968). MR 0230317 (37:5879)
  • [J] James, I.M.: The suspension triad of a sphere.
    Ann. Math. 63, 407-429 (1956). MR 0079263 (18:58f)
  • [K] Klein, J.R.: The dualizing spectrum of a topological group.
    Math. Annalen, 421-456 (2001). MR 1819876 (2001m:55037)
  • [L] Larmore, L.L.: Twisted cohomology theories and the single obstruction to lifting.
    Pacific J. Math. 41, 755-769 (1972). MR 0353315 (50:5799)
  • [Ma] Mazur, B.: Differential topology from the point of view of simple homotopy theory.
    Pub. Math. Inst. Ht. Étud. Scient., No. 15 (1963). MR 0161342 (28:4550)
  • [R-S] Rourke, C. P.; Sanderson, B.J.: $ \triangle $-sets. I. Homotopy theory.
    Quart. J. Math. Oxford Ser. 22, 321-338 (1971). MR 0300281 (45:9327)
  • [Sm] Smale, S.: The classification of immersions of spheres in Euclidean spaces.
    Ann. of Math. 69, 327-344 (1959). MR 0105117 (21:3862)
  • [Sta] Stallings, J. R.: Embedding homotopy types into manifolds.
    1965 unpublished paper (see http://math.berkeley.edu/$ \sim $stall for a TeXed version).
  • [Str] Strøm, A.: The homotopy category is a homotopy category.
    Arch. Math. 23, 435-441 (1972). MR 0321082 (47:9615)
  • [W1] Wall, C. T. C.: Classification problems in differential topology--IV. Thickenings.
    Topology 5, 73-94 (1966). MR 0192509 (33:734)
  • [W2] Wall, C. T. C.: Surgery on Compact Manifolds.
    (Mathematical Surveys and Monographs, Vol. 69).
    Amer. Math. Soc. (1999). MR 1687388 (2000a:57089)

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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
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.



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