Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

DOI10.4064/FM263-1-2024arXiv2002.02388MaRDI QIDQ6504211

Łukasz Patryk Michalak, Wacław Marzantowicz

Abstract: In this work we present a construction of correspondence between epimorphisms $varphicolonp{i}_1(M)o{F}_r$ from the fundamental group of a compact manifold $M$ onto the free group of rank $r$, and systems of $r$ framed non-separating hypersurfaces in $M$, which induces a bijection onto their framed cobordism classes. In consequence, for closed manifolds any such $varphi$ can be represented by the Reeb epimorphism of a Morse function $fcolonMomathbbR$, i.e. by the epimorphism induced by the quotient map $MomathcalR(f)$ onto the Reeb graph of $f$. Applying this construction we discuss the problem of classification up to (strong) equivalence of epimorphisms onto free groups providing a new purely geometrical-topological proof of the solution of this problem for surface groups.

Mathematics Subject Classification ID

Geometric group theory (20F65) Relations of low-dimensional topology with graph theory (57M15) Other types of cobordism (57R90)

This page was built for publication: Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6504211)