On realizing homology classes by maps of restricted complexity

TL;DR

The paper demonstrates that in higher codimensions, certain mod 2 homology classes cannot be realized by immersions or maps with limited singularities, providing explicit cohomological obstructions.

Contribution

It introduces explicit cohomology-based obstructions showing the non-realizability of some homology classes by restricted maps in high-dimensional manifolds.

Findings

Existence of non-realizable mod 2 homology classes in high dimensions.

Explicit cohomological obstructions for realizability.

Limitations on maps with restricted multi-singularities.

Abstract

We show that in every codimension greater than one there exists a mod 2 homology class in some closed manifold (of sufficiently high dimension) which 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'.