The moduli space of thickenings
Author:
Mokhtar Aouina
Journal:
Trans. Amer. Math. Soc. 364 (2012), 66896717
MSC (2010):
Primary 57R19; Secondary 55P10, 55P91, 55R25, 55Q99, 55S35, 55S40, 55U10, 55N99
Published electronically:
May 30, 2012
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Fix a finite connected CW complex of dimension . An thickening of is a pair in which is a compact dimensional manifold and 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 , and the connectivity of . 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 .
 [AK]
Aouina, M.; Klein, J.R.: On C.T.C. Wall's suspension theorem.
Forum Mathematicum 18, 829837 (2005).
 [B]
James
C. Becker, Extensions of cohomology theories, Illinois J.
Math. 14 (1970), 551–584. MR 0273598
(42 #8476)
 [CJ]
Michael
Crabb and Ioan
James, Fibrewise homotopy theory, Springer Monographs in
Mathematics, SpringerVerlag London, Ltd., London, 1998. MR 1646248
(99k:55001)
 [DDK]
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
(82a:55009), http://dx.doi.org/10.1090/S00029939198005879521
 [G]
Thomas
G. Goodwillie, Calculus. I. The first derivative of pseudoisotopy
theory, 𝐾Theory 4 (1990), no. 1,
1–27. MR
1076523 (92m:57027), http://dx.doi.org/10.1007/BF00534191
 [H]
Morris
W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 0119214
(22 #9980), http://dx.doi.org/10.1090/S00029947195901192144
 [Ho]
J.
P. E. Hodgson, Obstructions to concordance for thickenings,
Invent. Math. 5 (1968), 292–316. MR 0230317
(37 #5879)
 [J]
I.
M. James, The suspension triad of a sphere, Ann. of Math. (2)
63 (1956), 407–429. MR 0079263
(18,58f)
 [K]
John
R. Klein, The dualizing spectrum of a topological group, Math.
Ann. 319 (2001), no. 3, 421–456. MR 1819876
(2001m:55037), http://dx.doi.org/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
(50 #5799)
 [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
(28 #4550)
 [RS]
C.
P. Rourke and B.
J. Sanderson, Δsets. I. Homotopy theory, Quart. J.
Math. Oxford Ser. (2) 22 (1971), 321–338. MR 0300281
(45 #9327)
 [Sm]
Stephen
Smale, The classification of immersions of spheres in Euclidean
spaces, Ann. of Math. (2) 69 (1959), 327–344.
MR
0105117 (21 #3862)
 [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
(47 #9615)
 [W1]
C.
T. C. Wall, Classification problems in differential topology. IV.
Thickenings, Topology 5 (1966), 73–94. MR 0192509
(33 #734)
 [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 (2000a:57089)
 [AK]
 Aouina, M.; Klein, J.R.: On C.T.C. Wall's suspension theorem.
Forum Mathematicum 18, 829837 (2005).
 [B]
 Becker, J.C.: Extension of cohomology theories.
Illinois J. Math. 14, 551584 (1970). MR 0273598 (42:8476)
 [CJ]
 Crabb, M.; James, I.M.:
Fibrewise homotopy theory. Springer Monographs in Mathematics. SpringerVerlag London, 1998. MR 1646248 (99k:55001)
 [DDK]
 Dror, E.; Dwyer, W. G.; Kan, D. M.: Equivariant maps which are self homotopy equivalences.
Proc. Amer. Math. Soc. 80, no. 4, 670672 (1980). MR 587952 (82a:55009)
 [G]
 Goodwillie, T. G.: Calculus. I. The first derivative of pseudoisotopy theory.
Theory 4, no. 1, 127 (1990). MR 1076523 (92m:57027)
 [H]
 Hirsch, M.: Immersions of manifolds.
Trans. Amer. Math. Soc. 93, 242276 (1959). MR 0119214 (22:9980)
 [Ho]
 Hodgson, J.P.E.: Obstructions to concordance for thickenings.
Inventiones Math. 5, 292316 (1968). MR 0230317 (37:5879)
 [J]
 James, I.M.: The suspension triad of a sphere.
Ann. Math. 63, 407429 (1956). MR 0079263 (18:58f)
 [K]
 Klein, J.R.: The dualizing spectrum of a topological group.
Math. Annalen, 421456 (2001). MR 1819876 (2001m:55037)
 [L]
 Larmore, L.L.: Twisted cohomology theories and the single obstruction to lifting.
Pacific J. Math. 41, 755769 (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)
 [RS]
 Rourke, C. P.; Sanderson, B.J.: sets. I. Homotopy theory.
Quart. J. Math. Oxford Ser. 22, 321338 (1971). MR 0300281 (45:9327)
 [Sm]
 Smale, S.: The classification of immersions of spheres in Euclidean spaces.
Ann. of Math. 69, 327344 (1959). MR 0105117 (21:3862)
 [Sta]
 Stallings, J. R.: Embedding homotopy types into manifolds.
1965 unpublished paper (see http://math.berkeley.edu/stall for a TeXed version).
 [Str]
 Strøm, A.: The homotopy category is a homotopy category.
Arch. Math. 23, 435441 (1972). MR 0321082 (47:9615)
 [W1]
 Wall, C. T. C.: Classification problems in differential topologyIV. Thickenings.
Topology 5, 7394 (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)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
57R19,
55P10,
55P91,
55R25,
55Q99,
55S35,
55S40,
55U10,
55N99
Retrieve articles in all journals
with MSC (2010):
57R19,
55P10,
55P91,
55R25,
55Q99,
55S35,
55S40,
55U10,
55N99
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/S000299472012056455
PII:
S 00029947(2012)056455
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.
