A lower bound in the problem of realization of cycles

Abstract

AbstractWe consider the classical Steenrod problem on realization of integral homology classes by continuous images of smooth oriented manifolds. Let be the smallest positive integer such that any integral ‐dimensional homology class becomes realizable in the sense of Steenrod after multiplication by . The best known upper bound for was obtained independently by Brumfiel and Buchstaber in 1969. All known lower bounds for were very far from this upper bound. The main result of this paper is a new lower bound for that is asymptotically equivalent to the Brumfiel–Buchstaber upper bound (in the logarithmic scale). For , we prove that our lower bound is exact. Also, we obtain analogous results for the case of realization of integral homology classes by continuous images of smooth stably complex manifolds.