Actions of finite \(p\)-groups on homology manifolds (Q2758158)

One of the basic problems in the theory of transformation groups is to study the relation between \(G\)-spaces and the fixed point sets. When a \(G\)-space is a space with some quite nice properties, many basic and famous results have been obtained, such as Smith's theorem, the Atiyah-Singer \(G\)-signature theorem and so on.NEWLINENEWLINEThe author of this paper considers orientable \({\mathbb Z}_{(p)}\)-homology manifolds with simplicial finite \(p\)-group actions. He first gives a relation between the Witt classes associated to the \({\mathbb F}_p\)-cohomology rings of a \({\mathbb Z}_{(p)}\)-Poincaré duality space \(X\) with a \({\mathbb Z}_p\)-action and its fixed point set, where \(p\) is an odd prime. This is a generalization of the result obtained by \textit{J. P. Alexander} and \textit{G. C. Hamrick} [Comment. Math. Helv. 53, 149--159 (1978; Zbl 0424.18014)], where \(X\) is weakened to a \({\mathbb Z}_{(p)}\)-Poincaré duality space from an integral Poincaré duality space.NEWLINENEWLINEFurthermore, the author obtains an interesting Witt class comparison result for actions of finite \(p\)-groups on \({\mathbb Z}_{(p)}\)-homology manifolds as follows: Let \(X\) be an oriented \(2n\)-dimensional \({\mathbb Z}_{(p)}\)-homology manifold. Suppose that \(p\geq \dim_{{\mathbb F}_p}H^*(X;{\mathbb F}_p)+1\) and let a finite \(p\)-group \(G\) act simplicially on \(X\) with appropriate orientations of the fixed point set components of \(X\). Then NEWLINE\[NEWLINEw(H^{\text{ev}}(X;{\mathbb F}_p))=\sum_{F\subset X^G}w(H^{\text{ev}}(F;{\mathbb F}_p))NEWLINE\]NEWLINE in \(W({\mathbb F}_p)\).