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 En(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 $$k(X) = \sum\limits_{{\text{ }}r \equiv 0{\text{ }}({\text{mod 2}})} {{\text{ dim }}H_r (X;\mathbb{Q}){\text{ mod 2 }} \in {\text{ }}\mathbb{Z}_2 .} $$ Bibliography: 7 titles.