On realizing homology classes by maps of restricted complexity

Abstract

We show that in every codimension greater than 1, there exists a mod 2 homology class in some closed manifold (of sufficiently high dimension) that cannot be realized by an immersion of closed manifolds. The proof gives explicit obstructions (in terms of cohomology operations) for realizability of mod 2 homology classes by immersions. We also prove the corresponding result in which the word 'immersion' is replaced by 'map with some restricted set of multi-singularities'.Theorem 1.2. Let k > 1. Let I be an admissible sequence of excess e(I) = k, and let Sq I ∈ A be the corresponding monomial. If the cohomology class x ∈ H k (N ; Z 2 ) is realizable by an immersion, then Sq I (x) is the reduction mod 2 of an integral class.In particular, if k is even and β(x 2 ) is nonzero (where β is the Bockstein associated to reduction mod 2), then x cannot be realized by an immersion.

Homologies are infinitely complex

“…In the present paper we show that stratified sets of finite complexity are never sufficient to realize all homology classes of codimension k > 1 (Theorem 2.4). The proof is almost identical to that of Theorem 1.3 in [1].…”

“…Proof. In [1,Proposition 4.2] it was shown that if f : M m → P p is a generic smooth map, p = m + k, and α ∈ H k (P ; Z 2 ) the cohomology class realized by f , then the class β(α 2 ) ∈ H 2k+1 H(P ; Z) (where β is the Bockstein operator) is dual to the homology class f * Σ(f ) , where Σ(f ) is the set of singular points. By Remark 5.2, for a dc map the class β(α 2 ) must be zero.…”

“…(3) In [1] it was shown that for any k > 1 there are k-dimensional cohomology classes of some sufficiently high-dimensional manifolds which cannot be realized by immersions. (4) In [1] we have shown also that if we allow only a finite set of multisingularities, then not all cohomology classes can be realized.…”

“…(3) In [1] it was shown that for any k > 1 there are k-dimensional cohomology classes of some sufficiently high-dimensional manifolds which cannot be realized by immersions. (4) In [1] we have shown also that if we allow only a finite set of multisingularities, then not all cohomology classes can be realized. In the present paper we show that stratified sets of finite complexity are never sufficient to realize all homology classes of codimension k > 1 (Theorem 2.4).…”

On the Hodge theory of stratified spaces

“…(Thom's results do not hold if we impose conditions on f beyond being continuous, e.g., [55] shows that there are homology classes with Z 2 coefficients that can not be realized by immersed submanifolds. See also [22,29,99,101].…”