The Lefschetz fixed point theorem for compact groups

Author:
Ronald J. Knill

Journal:
Proc. Amer. Math. Soc. **66** (1977), 148-152

MSC:
Primary 55C20; Secondary 22C05

MathSciNet review:
0454962

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Abstract: It is shown that every compact group *G* is a *Q*-simplicial space where *Q* is any field of characteristic zero. As a consequence it follows that *G* satisfies a variation of the Lefschetz fixed point theorem.

It has been known for some time that the Lefschetz fixed point theorem applies to a few spaces other than just ANR spaces, especially if some care is taken to use coefficients in certain fields [2]. The case of all compact groups provides a broad class of spaces which may not have local connectivity of any order. It is shown that every compact group *G* satisfies the Lefschetz fixed point theorem when coefficients for the homology groups are taken in a field of characteristic zero.

**[1]**C. H. Dowker,*Homology groups of relations*, Ann. of Math. (2)**56**(1952), 84–95. MR**0048030****[2]**R. J. Knill,*𝑄-simplicial spaces*, Illinois J. Math.**14**(1970), 40–51. MR**0258017****[3]**Solomon Lefschetz,*Algebraic Topology*, American Mathematical Society Colloquium Publications, v. 27, American Mathematical Society, New York, 1942. MR**0007093****[4]**André Weil,*L’intégration dans les groupes topologiques et ses applications*, Actual. Sci. Ind., no. 869, Hermann et Cie., Paris, 1940 (French). [This book has been republished by the author at Princeton, N. J., 1941.]. MR**0005741**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1977-0454962-4

Keywords:
Compact group,
Lefschetz fixed point theorem,
*Q*-simplicial spaces

Article copyright:
© Copyright 1977
American Mathematical Society