Quadratic Property of the Rational Semicharacteristic

Abstract

Let $n \equiv 1(\bmod 4)$. Assume that V is a manifold, $E_n (V)$ is the set of germs of n-dimensional oriented submanifolds of V, and $!E_n (V)$ is the ℤ₂-module of all ℤ₂-valued functions on E _n (V). If $X^n \subset V$ is an oriented submanifold, let $1_x \in !E_n (V)$ be the indicator function of the set of germs of X. It is proved that there exists a quadratic map $q:!E_n (V) \to \mathbb{Z}_2$ such that for any compact oriented submanifold $X^{{\text{ }}n} \subset V$ one has the relation $q(1_X ) = k(X)$, where $k(X)$ is the (rational) semicharacteristic of $X^{{\text{ }}n}$, i.e., the residue class defined by the formula

$$k(X) = \sum\limits_{{\text{ }}r \equiv 0{\text{ }}({\text{mod 2}})} {{\text{ dim }}H_r (X;\mathbb{Q}){\text{ mod 2 }} \in {\text{ }}\mathbb{Z}_2 .} $$

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