Domination of manifolds by hypersurfaces
In this short note we prove that any smooth, closed, oriented manifold can be dominated by a codimension 1 submanifold of the sphere.
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arXiv subjects
Vasilii Rozhdestvenskii
Publications and source records attributed to Vasilii Rozhdestvenskii.
Highly divisible cycles in homological Atiyah-Hirzebruch spectral sequences
Let $F_*$ be a homology theory of finite type and $\{E^r_{s,t}(X)=\frac{Z^r_{s,t}(X)}{B^r_{s,t}(X)}; d^r_{s,t}\}$ denote the Atiyah-Hirzebruch spectral sequence for $F_*$ of a $CW$-complex $X$. In this paper we study elements of the maximal possible order in the groups $E^2_{n,0}(X)/Z^{r}_{n,0}(X)$.
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A lower bound in the problem of realization of cycles
We consider the classical Steenrod problem on realization of integral homology classes by continuous images of smooth oriented manifolds. Let $k(n)$ be the smallest positive integer such that any integral $n$-dimensional homology class becomes realizable in the sense of Steenrod after multiplication by $k(n)$. The best known upper bound for $k(n)$ was obtained independently by G. Brumfiel and V. Buchstaber in 1969. All known lower bounds for $k(n)$ were very far from this upper bound. The main result of this paper is a new lower bound for $k(n)$ which is asymptotically equivalent to the Brumfiel-Buchstaber upper bound (in the logarithmic scale). For $n<24$ 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.
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