Higher dimensional 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 . We show that for n > 3, if cat S 1 (M n ) = 2 then M n ≈ S n or M n ≈ S n−1 × S 1 or the non-orientable S n−1 -bundle over S 1 . 1 2

Old and new categorical invariants of manifolds

“…Since H i (Y k ) = 0 for i > 1 it follows that t m − 1 : H i (Ỹ ) → H i (Ỹ ) is an automorphism for each m ≥ 1. An algebraic lemma (Lemma 4 of [19]) then implies thatH i (Ỹ ) = 0 for all i and soỸ is contractible and Y is a homotopy S 1 . By Stallings [46] and Freedman and Quinn [10], n−2 1 is a topologically trivial knot in S n 1 and it follows that M n = M n 1 ≈ S n−1 ×S 1 .…”

“…Finally, it is shown in [19] that 1(c) For n > 3, cat S 1 (M n ) = 2 and π 1 (M n ) = Z implies M is homeomorphic to an S n−1 -bundle over S 1 .…”