Isovariant homotopy equivalences of manifolds with group actions

Author:
Reinhard Schultz

Journal:
Proc. Amer. Math. Soc. **144** (2016), 1363-1370

MSC (2010):
Primary 55P91, 57S17; Secondary 55R91

DOI:
https://doi.org/10.1090/proc/12795

Published electronically:
August 11, 2015

MathSciNet review:
3447686

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Abstract: Let be an equivariant homotopy equivalence of connected closed manifolds with smooth semifree actions of a finite group , and assume also that is isovariant. The main result states that is a homotopy equivalence in the category of isovariant mappings if the manifolds satisfy a Codimension Gap Hypothesis; this is done by showing directly that satisfies the criteria in the Isovariant Whitehead Theorem of G. Dula and the author. Examples are given to show the need for the hypotheses in the main result.

**[1]**Aaron Christian Beshears,*G-isovariant structure sets and stratified structure sets*, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)-Vanderbilt University. MR**2695560****[2]**Glen E. Bredon,*Introduction to compact transformation groups*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR**0413144 (54 #1265)****[3]**Glen E. Bredon,*Sheaf theory*, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR**1481706 (98g:55005)**- [4]
W. Browder,
*Isovariant homotopy equivalence*, Abstracts Amer. Math. Soc.**8**(1987), 237-238. **[5]**William Browder and Frank Quinn,*A surgery theory for -manifolds and stratified sets*, Manifolds--Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 27-36. MR**0375348 (51 #11543)****[6]**Sylvain E. Cappell,*A splitting theorem for manifolds*, Invent. Math.**33**(1976), no. 2, 69-170. MR**0438359 (55 #11274)****[7]**Michael Davis,*Smooth -manifolds as collections of fiber bundles*, Pacific J. Math.**77**(1978), no. 2, 315-363. MR**510928 (80b:57034)****[8]**Giora Dula and Reinhard Schultz,*Diagram cohomology and isovariant homotopy theory*, Mem. Amer. Math. Soc.**110**(1994), no. 527, viii+82. MR**1209409 (95a:55028)**, https://doi.org/10.1090/memo/0527**[9]**André Haefliger and Valentin Poenaru,*La classification des immersions combinatoires*, Inst. Hautes Études Sci. Publ. Math.**23**(1964), 75-91 (French). MR**0172296 (30 #2515)****[10]**Soren Arnold Illman,*Equivariant algebraic topology*, ProQuest LLC, Ann Arbor, MI, 1972. Thesis (Ph.D.)-Princeton University. MR**2622205****[11]**Sören Illman,*Smooth equivariant triangulations of -manifolds for a finite group*, Math. Ann.**233**(1978), no. 3, 199-220. MR**0500993 (58 #18474)****[12]**Takao Matumoto,*On - complexes and a theorem of J. H. C. Whitehead*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**18**(1971), 363-374. MR**0345103 (49 #9842)****[13]**Richard S. Palais,*The classification of -spaces*, Mem. Amer. Math. Soc. No. 36, 1960. MR**0177401 (31 #1664)****[14]**Reinhard Schultz,*Differentiable group actions on homotopy spheres. I. Differential structure and the knot invariant*, Invent. Math.**31**(1975), no. 2, 105-128. MR**0405471 (53 #9264)****[15]**Reinhard Schultz,*Isovariant homotopy theory and differentiable group actions*, Algebra and topology 1992 (Taejŏn), Korea Adv. Inst. Sci. Tech., Taejŏn, 1992, pp. 81-148. MR**1212981 (94i:57056)****[16]**Sandor Howard Straus,*Equivariant codimension one surgery*, ProQuest LLC, Ann Arbor, MI, 1972. Thesis (Ph.D.)-University of California, Berkeley. MR**2940154****[17]**D. W. Sumners,*Smooth -actions on spheres which leave knots pointwise fixed*, Trans. Amer. Math. Soc.**205**(1975), 193-203. MR**0372893 (51 #9097)****[18]**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)****[19]**Shmuel Weinberger,*The topological classification of stratified spaces*, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994. MR**1308714 (96b:57024)****[20]**George W. Whitehead,*Elements of homotopy theory*, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR**516508 (80b:55001)**

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Additional Information

**Reinhard Schultz**

Affiliation:
Department of Mathematics, University of California at Riverside, Riverside, California 92521

Email:
schultz@math.ucr.edu

DOI:
https://doi.org/10.1090/proc/12795

Received by editor(s):
January 20, 2015

Received by editor(s) in revised form:
March 17, 2015

Published electronically:
August 11, 2015

Communicated by:
Michael A. Mandell

Article copyright:
© Copyright 2015
American Mathematical Society