Quadratic property of the rational semicharacteristic

S. S. Podkorytov

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 5, Tome 267 (2000), pp. 241-259

Résumé

Let $n\equiv1\pmod4$. Let $V$ be a manifold, $\mathbf E_n(V)$ the set of germs of $n$-dimensional oriented submanifolds of $V$, and $!\mathbf E_n(V)$ the $\mathbb Z_2$-module of all $\mathbb Z_2$-valued functions on $\mathbf E_n(V)$. For a oriented submanifold $X^n\subset V$ let $\mathbf1(X)\in!\mathbf E_n(V)$ be the indicator function of the set of germs of $X$. It is proved that there exists a quadratic map $q\colon!\mathbf E_n(V)\to\mathbb Z_2$ such that for any compact oriented submanifold $X^n\subset V$ one has the relation $q(\mathbf1(X))=\textrm{к}(X)$, where $\textrm{к}(X)$ is the (rational)semicharacteristic of $X^n$, i.e., the residue class defined by the formula $$ \textrm{к}(X)=\sum_{r\equiv0\pmod2}\dim H_r(X;\mathbb Q)\bmod2\in\mathbb Z_2. $$