Semicharacteristics of oriented Grassmannians
Let M be a closed manifold of dimension \(2m+1\). The semicharacteristic of M with respect to a field F is defined to be \[ S\chi_ F(M)=\sum^{m}_{0}(-1)^ i\quad \dim_ F H^ i(M;F) mod 2. \] If F is of characteristic not 2, then one assumes that M is oriented. In this paper the author proves that the semicharacteristics of \(G=G^+_ k({\mathbb{R}}^{n+k})\), the oriented Grassmannians, for nk odd are independent of the field and given by \(\left( \begin{matrix} [n]+[k]\\ [k]\end{matrix} \right)\) (mod 2). For nk\(\equiv 1\) mod 4 the result is deduced from a theorem of \textit{G. Lusztig, J. Milnor} and \textit{F. P. Peterson} [Topology 8, 357-359 (1969; Zbl 0165.263)], and by examing the cohomology ring of G. For the case nk\(\equiv 3\) mod 4 the result is proved by constructing a cobordism (V,T) between a copy of G and a projective space bundle whose base is oriented Grassmannians of dimension \(\equiv 1\) mod 4 and by applying his formula \(S\chi (\partial V)=\chi (F)+\chi (F\cap F)\), where F is the fixed point set of an involution T and \(F\cap F\) is the self-intersection of F in V.
scientific article; zbMATH DE number 3915244
scientific article