User scott van thuong - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T05:13:27Z http://mathoverflow.net/feeds/user/17233 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73159/does-a-fixed-point-free-homotopy-involution-imply-that-a-manifold-bounds Does a fixed-point free "homotopy involution" imply that a manifold bounds? Scott Van Thuong 2011-08-18T15:08:46Z 2011-08-18T19:19:57Z <p>Let $M^n$ be a closed (compact, connected, without boundary) smooth manifold. It is known that if there exists a fixed point free involution $\tau:M \rightarrow M$, then M bounds. That is, there exists a compact manifold $W^{n+1}$ such that $\partial W = M$.</p> <p>But now suppose $\tau$ is only a "homotopy involution". That is $\tau^2$ is only homotopic to the identity on $M$ rather than equal to the identity. Can we say that $M$ bounds?</p> <p>For some reason I feel this statement is not true..., but I have not been able to construct a counterexample yet. For a counterexample, maybe an aspherical, nonbounding manifold would be the best candidate?</p> <p>On a related question, what if we say that $\tau^2$ is <em>isotopic</em> to the identity on M. Then does M bound?</p> <p>Thanks, I appreciate any responses.</p>