Continuous maps of manifolds with involution. III
Minoru Nakaoka · Osaka City University (Osaka City University) · 1974
IntroductionLet N and M be m-dimensional closed manifolds on each of which an involution T is given, and )etf:N-+M be a continuous map.In the preceding paper [7], on the assumption that the involutions T of M and iV are both free the author introduced a mod 2 integer %(/) called the equivariant Lefschetz number of/, and proved that if %(/)^0 then/has an equivariant point.In this paper the result will be generalized to the case when the involution T of N is not necessarily free.The former result was proved through the use of the equivariant point index /(/), which is constructed from the class Δ oo ^Hm (S°°xM 2 ) requiring that T the involution T of N is free (see [7]).Taking in place of S°°χM 2 the pair of T the symmetric product of M and its diagonal, we define a new equivariant point index /(/) provided that the involution of M is free and the involution of N is non-trivial.The new result will be proved by making use of the new index.Recently, to show that certain homotopy classes in closed manifolds cannot be realized by embedded sphere, R. Fenn [5] has proved a theorem of the Borsuk-Ulam type.In this paper, /(/) will be also used to generalize the Fenn theorem.Throughout this paper, the homology and cohomology with coefficients in Z 2 are to be understood.