|
Borsuk-Ulam type theorems for manifolds
Author:
Oleg R. Musin
Journal:
Proc. Amer. Math. Soc. 140 (2012), 2551-2560
MSC (2010):
Primary 55M35, 55M99, 57R85
Posted:
November 18, 2011
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: This paper establishes a Borsuk-Ulam type theorem for PL-
manifolds with a finite group action, depending on the free equivariant cobordism class of a manifold. In particular, necessary and sufficient conditions are considered for a manifold with a free involution to be of Borsuk-Ulam type.
References
- 1.
Imre
Bárány, Borsuk’s theorem through complementary
pivoting, Math. Programming 18 (1980), no. 1,
84–88. MR
557116 (81d:90099), http://dx.doi.org/10.1007/BF01588299
- 2.
Edward
Bierstone, General position of equivariant maps, Trans. Amer.
Math. Soc. 234 (1977), no. 2, 447–466. MR 0464287
(57 #4221)
- 3.
Glen
E. Bredon, Introduction to compact transformation groups,
Academic Press, New York, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
(54 #1265)
- 4.
P.
E. Conner and E.
E. Floyd, Fixed point free involutions and equivariant maps,
Bull. Amer. Math. Soc. 66 (1960), 416–441. MR 0163310
(29 #613)
- 5.
P.
E. Conner and E.
E. Floyd, Differentiable periodic maps, Ergebnisse der
Mathematik und ihrer Grenzgebiete, N. F., Band 33, Academic Press Inc.,
Publishers, New York, 1964. MR 0176478
(31 #750)
- 6.
M.
J. Field, Transversality in 𝐺-manifolds, Trans. Amer.
Math. Soc. 231 (1977), no. 2, 429–450. MR 0451276
(56 #9563)
- 7.
D.
B. Fuks and V.
A. Rokhlin, Beginner’s course in topology, Universitext,
Springer-Verlag, Berlin, 1984. Geometric chapters; Translated from the
Russian by A. Iacob; Springer Series in Soviet Mathematics. MR 759162
(86a:57001)
- 8.
Sören
Illman, Smooth equivariant triangulations of 𝐺-manifolds
for 𝐺 a finite group, Math. Ann. 233 (1978),
no. 3, 199–220. MR 0500993
(58 #18474)
- 9.
Jiří
Matoušek, Using the Borsuk-Ulam theorem, Universitext,
Springer-Verlag, Berlin, 2003. Lectures on topological methods in
combinatorics and geometry; Written in cooperation with Anders Björner
and Günter M. Ziegler. MR 1988723
(2004i:55001)
- 10.
Mark
D. Meyerson and Alden
H. Wright, A new and constructive proof of the
Borsuk-Ulam theorem, Proc. Amer. Math. Soc.
73 (1979), no. 1,
134–136. MR
512075 (80e:55006), http://dx.doi.org/10.1090/S0002-9939-1979-0512075-9
- 11.
O. R. Musin, Sperner and Tucker's lemmas and its generalizations, in preparation.
- 12.
Dušan
Repovš and Arkady
Skopenkov, On projected embeddings and desuspending the
𝛼-invariant, Topology Appl. 124 (2002),
no. 1, 69–75. MR 1926135
(2003g:57046), http://dx.doi.org/10.1016/S0166-8641(01)00237-1
- 13.
H.
Steinlein, Borsuk’s antipodal theorem and its generalizations
and applications: a survey, Topological methods in nonlinear analysis,
Sém. Math. Sup., vol. 95, Presses Univ. Montréal,
Montreal, QC, 1985, pp. 166–235. MR 801938
(86k:55002)
- 14.
A.
S. Švarc, The genus of a fibered space, Trudy Moskov.
Mat. Obšč. 10 (1961), 217–272
(Russian). MR
0154284 (27 #4233)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
55M35,
55M99,
57R85
Retrieve articles in all journals
with MSC (2010):
55M35,
55M99,
57R85
Additional Information
Oleg R. Musin
Affiliation:
Department of Mathematics, University of Texas at Brownsville, 80 Fort Brown, Brownsville, Texas 78520
Email:
oleg.musin@utb.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11094-3
PII:
S 0002-9939(2011)11094-3
Keywords:
Borsuk-Ulam theorem,
group action,
equivariant cobordism.
Received by editor(s):
December 4, 2009
Received by editor(s) in revised form:
May 24, 2010; December 18, 2010; January 13, 2011; January 18, 2011; January 23, 2011; and February 24, 2011
Posted:
November 18, 2011
Additional Notes:
This research was supported in part by NSF grant DMS-0807640 and NSA grant MSPF-08G-201.
Communicated by:
Alexander N. Dranishnikov
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
|
