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On the Stiefel-Whitney numbers and the semicharacteristic. (English. Russian original) Zbl 1039.57019

St. Petersbg. Math. J. 14, No. 5, 835-846 (2003); translation from Algebra Anal. 14, No. 5, 171-187 (2002).
The aim of this paper is to prove the following two theorems (all manifolds and submanifolds mentioned in this review are smooth and without boundary).
Theorem 1. For fixed non-negative integers \(m\) and \(q\) such that \(m\leq q\), let \(E_ m^ q\) denote the set of germs of \(m\)-dimensional submanifolds of \(\mathbb R^ q\), and let \(\mathcal N_m\) be the \(m\)-dimensional unoriented cobordism group, considered as a \(\mathbb Z_2\)-vector space. Then there exists a linear mapping \(B\) from the \(\mathbb Z_2\)-vector space of all functions \(E_ m^ q\rightarrow \mathbb Z_2\) to \(\mathcal N_m\) such that, for each compact \(m\)-dimensional submanifold \(X\subset \mathbb R^ q\), the value \(B(I(X))\) is the cobordism class of \(X\), if \(I(X)\) is the indicator function of the set of those germs associated to \(X\).
Before stating the second theorem, denote by \(\gamma_ m^ q\) the canonical \(m\)-plane bundle over the Grassmann manifold \(G_ m^ q\) of all \(m\)-dimensional vector subspaces in \(\mathbb R^q\). Given a topological space \(B\) and a continuous mapping \(f:B\rightarrow G_ m^ q\), call a pair \((X,g)\) a compact \(m\)-dimensional \(f\)-submanifold in \(\mathbb R^ q\), if \(X\subset \mathbb R^q\) is a compact \(m\)-dimensional submanifold and \(g: X\rightarrow B\) is a continuous mapping such that the composite \(f\circ g: X\rightarrow G_ m^ q\) assigns to each \(x\in X\) the tangent space of \(X\) at \(x\). Finally, denote by \(E_ f\) the set of germs of such \(f\)-submanifolds.
Theorem 2. Fix non-negative integers \(m\), \(n\), \(q\) such that \(m+1=2n\) and \(m\leq q\), a topological space \(B\), and a continuous mapping \(f:B\rightarrow G_ m^ q\) such that the homomorphism induced in \(\mathbb Z_2\)-cohomology, \(f^ \ast:H^\ast (G_ m^ q) \rightarrow H^\ast (B)\), maps the Wu class \(v_ n (\gamma_ m^ q)\in H^ n (G_ m^ q)\) to zero. Then there exists a quadratic functional \(K\) from the \(\mathbb Z_2\)-vector space of all functions \(E_ f \rightarrow \mathbb Z_2\) to \(\mathbb Z_2\) such that, for each compact \(m\)-dimensional \(f\)-submanifold \((X,g)\) in \(\mathbb R^ q\), the value \(K(I(X,g))\) is the Kervaire \(\mathbb Z_ 2\)-semicharacteristic of \(X\), if \(I(X,g)\) is the indicator function of the set of those germs associated to \((X,g)\).

MSC:

57R20 Characteristic classes and numbers in differential topology
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
57R19 Algebraic topology on manifolds and differential topology
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