Fundamental groups of manifolds with S 1-category 2

Abstract

A closed topological n-manifold M n is of S 1 -category 2 if it can be covered by two open subsets W1,W2 such that the inclusions Wi → M n factor homotopically through maps Wi → S 1 → M n . We show that the fundamental group of such an n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation. In particular, if n = 3, the fundamental group is cyclic.

Higher dimensional manifolds with S 1-category 2

“…The following proposition, proved in [6], allows us to replace the open sets W i by compact submanifolds that meet only along their boundaries.…”

“…In [6] it is shown that for n = 2, 3 these are the only possibilities up to homotopy type, hence by Perelman [15] this is true up to homeomorphism type. For n > 3 it is shown in [7] that π 1 (M n ) is trivial or infinite cyclic.…”

Manifolds with S 1-category 2 have cyclic fundamental groups

“…By results of Olum [10] and Perelman [9] this implies that cat S 1 M 3 = 2 if and only if M 3 is a lens space or M 3 is the non-orientable S 2 -bundle over S 1 . For the case n > 3 we showed [6] that cat S 1 M n = 2 implies that π 1 (M n ) is cyclic or a nontrivial product with amalgamation A * C B of cyclic groups. In this paper we show that in this case π 1 (M n ) is in fact cyclic (Corollaries 1 and 2):…”

“…In [6] it was shown that for a closed 3-manifold M 3 we have cat S 1 M 3 = 2 if and only if π 1 (M 3 ) is cyclic. By results of Olum [10] and Perelman [9] this implies that cat S 1 M 3 = 2 if and only if M 3 is a lens space or M 3 is the non-orientable S 2 -bundle over S 1 .…”

Coverings of 3-Manifolds by Three Open Solid Tori

“…Thus when A is a point P , cat P (M n ) = cat(M n ). In the case A =S 1 it was shown in [GGH2] that the fundamental group of a closed 3-manifold M with cat S 1 (M ) = 2 is cyclic and it then follows from Perelman's work [MT] that in this case M is a lens space; hence M can be covered by two open solid tori. As a first step to obtaining a list of all 3-manifolds with cat S 1 (M ) = 3 we ask about minimal covers of M by three open sets, each homotopy equivalent to S 1 .…”