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International conference "GEOMETRY, TOPOLOGY, ALGEBRA and NUMBER THEORY, APPLICATIONS" dedicated to the 120th anniversary of Boris Delone (1890–1980)
August 20, 2010 10:00, Moscow

Sets of links of vertices of triangulated manifolds and combinatorial approach to Steenrod's problem on realisation of cycles

Alexander Gaifullin
 Video records: Windows Media 313.2 Mb Flash Video 642.8 Mb MP4 642.8 Mb

Further, we are going to discuss an application of this construction to N. Steenrod's problem on realisation of cycles. It is well known that according to a result of R. Thom, any $n$-dimensional integral homology class $z$ of any topological space $X$ can be realised with some multiplicity by an image of an oriented smooth closed manifold $N^n$. Our new approach is based on an explicit combinatorial procedure for resolving singularities of a cycle. We give an explicit combinatorial construction that, for a given homology class $z$, yields a manifold $N^n$ and its mapping to $X$ which realises the class $z$ with some multiplicity. Moreover, the obtained manifold $N^n$ appears to be a finite-fold non-ramified covering over a very interesting special manifold $M^n$, which can be regarded either as an isospectral manifold of symmetric tridiagonal real $(n+1)\times(n+1)$-matrices or as a small covering over a permutohedron.