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 a finite connected CW complex of dimension . An *-thickening* of is a pair

In this paper we remove the connectivity hypothesis on . We define *moduli space* of -thickenings . We also define a *suspension map*

and compute its homotopy fibers in a range depending only on and . 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 .

**[A-K]**Aouina, M.; Klein, J.R.: On C.T.C. Wall's suspension theorem.*Forum Mathematicum***18**, 829-837 (2005).**[B]**James C. Becker,*Extensions of cohomology theories*, Illinois J. Math.**14**(1970), 551–584. MR**0273598****[C-J]**Michael Crabb and Ioan James,*Fibrewise homotopy theory*, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1998. MR**1646248****[D-D-K]**E. Dror, W. G. Dwyer, and D. M. Kan,*Equivariant maps which are self homotopy equivalences*, Proc. Amer. Math. Soc.**80**(1980), no. 4, 670–672. MR**587952**, 10.1090/S0002-9939-1980-0587952-1**[G]**Thomas G. Goodwillie,*Calculus. I. The first derivative of pseudoisotopy theory*, 𝐾-Theory**4**(1990), no. 1, 1–27. MR**1076523**, 10.1007/BF00534191**[H]**Morris W. Hirsch,*Immersions of manifolds*, Trans. Amer. Math. Soc.**93**(1959), 242–276. MR**0119214**, 10.1090/S0002-9947-1959-0119214-4**[Ho]**J. P. E. Hodgson,*Obstructions to concordance for thickenings*, Invent. Math.**5**(1968), 292–316. MR**0230317****[J]**I. M. James,*The suspension triad of a sphere*, Ann. of Math. (2)**63**(1956), 407–429. MR**0079263****[K]**John R. Klein,*The dualizing spectrum of a topological group*, Math. Ann.**319**(2001), no. 3, 421–456. MR**1819876**, 10.1007/PL00004441**[L]**Lawrence L. Larmore,*Twisted cohomology theories and the single obstruction to lifting*, Pacific J. Math.**41**(1972), 755–769. MR**0353315****[Ma]**Barry Mazur,*Differential topology from the point of view of simple homotopy theory*, Inst. Hautes Études Sci. Publ. Math.**15**(1963), 93. MR**0161342****[R-S]**C. P. Rourke and B. J. Sanderson,*Δ-sets. I. Homotopy theory*, Quart. J. Math. Oxford Ser. (2)**22**(1971), 321–338. MR**0300281****[Sm]**Stephen Smale,*The classification of immersions of spheres in Euclidean spaces*, Ann. of Math. (2)**69**(1959), 327–344. MR**0105117****[Sta]**Stallings, J. R.: Embedding homotopy types into manifolds.

1965 unpublished paper (see http://math.berkeley.edu/stall for a TeXed version).**[Str]**Arne Strøm,*The homotopy category is a homotopy category*, Arch. Math. (Basel)**23**(1972), 435–441. MR**0321082****[W1]**C. T. C. Wall,*Classification problems in differential topology. IV. Thickenings*, Topology**5**(1966), 73–94. MR**0192509****[W2]**C. T. C. Wall,*Surgery on compact manifolds*, 2nd ed., Mathematical Surveys and Monographs, vol. 69, American Mathematical Society, Providence, RI, 1999. Edited and with a foreword by A. A. Ranicki. MR**1687388**

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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:
https://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.