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Quadratic Property of the Rational Semicharacteristic

Authors:

S S Podkorytov

Institution:

(1) St.Petersburg Department of the, Steklov Mathematical Institute, Russia

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

₂-module of all Zopf

₂-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

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.

Keywords:

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