Abstract:
To define the homology groups of a space one needs to point out what is supposed to be a cycle and which cycles are supposed to be homologous. Initially, Poincaré defined a cycle in a manifold to be a smooth submanifold without boundary. Later he came to the definition of a cycle as an algebraic sum of singular simplices. This led to the singular homology theory. In the middle of the 20th century it was understood that the definition of a cycle as a smooth manifold leads to another important homology theory, which was called the bordism theory. Naturally, the question about the relationship between the two definitions of a cycle was posed. In particular, Steenrod in the late 1940s asked the following question.
Let $z$ be an integral homology class of a space $X$. Is it possible to realise $z$ as the image of the fundamental class of an oriented smooth manifold $M^n$ under a continuous mapping $f\colon M^n\to X$? In 1954 Thom constructed a non-realisable 7-dimensional class. However, he proved that for every dimension $n$, there exists a postitive integer $k(n)$ such that the class $k(n)z$ is always realisable.
In the talk we shall show how, given a singular cycle, to construct a manifold $M^n$ and a mapping $f\colon M^n\to X$ that realise the homology class of the given cycle with a certain multiplicity. This is based on a group-theoretic construction involving right-angular Coxeter groups. The construction of explicit realisation of cycles was obtained by the speaker in 2007, but its description on the language of Coxeter groups is new. In every dimension $n$, this construction allows one to point out a manifold $M_0^n$ that has the following universal property:
($*$) For each $X$ and each $z\in H_n(X,\mathbb Z)$ a multiple of $z$ can be realised as an image of the fundamental class of a finite-sheeted covering of $M^n_0$. We shall describe several properties of the class of all manifolds $M^n_0$ that have property ($*$). We shall find many examples of such manifolds among so-called small covers of simple polytopes. In dimension 4, we shall prove the Kotschick–Löh conjecture on realisation of cycles by hyperbolic manifolds.