Homology Spheres, Acyclic Groups and Kan-Thurston Theorem
Talk, Conference 'From Analysis to Homotopy Theory' , Greifswald, Germany
The theory of non-simply connected manifolds, the well-known area of algebraic topology, is closely related to combinatorial group theory and group cohomology. It is well known that for any finitely presented group $G$ and $n \geqslant 4$, there exists a smooth closed n-manifold whose fundamental group is isomorphic to $G$. This observation motivates the following question. Let us fix the class $\mathcal{C}$ of manifolds. What finitely presented groups can be fundamental groups of manifolds from that given class $\mathcal{C}$? The question is of interest in the case of $\mathcal{C}$ being the class of homology $n$-spheres, i.e., smooth closed $n$-manifolds with the homology of $n$-sphere. We review remarkable connections appearing in the study of homology spheres: we discuss applications of acyclic groups, the Kan-Thurston theorem, and smooth structures on spheres. We also present related author’s results. The site of the conference.