Does a fixed-point free "homotopy involution" imply that a manifold bounds? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T00:43:56Z http://mathoverflow.net/feeds/question/73159 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> http://mathoverflow.net/questions/73159/does-a-fixed-point-free-homotopy-involution-imply-that-a-manifold-bounds/73176#73176 Answer by Bruno Martelli for Does a fixed-point free "homotopy involution" imply that a manifold bounds? Bruno Martelli 2011-08-18T19:19:57Z 2011-08-18T19:19:57Z <p>A manifold with zero Euler characteristic admits a nowhere-vanishing vector field, which generates a one-parameter group of diffeomorphisms that are (smoothly) isotopic to the identity. A sufficiently small element $\tau$ is fixed-point free since the vector field does not vanish and the manifold is compact.</p> <p>There are manifolds with zero Euler characteristic that do not bound, for instance the unoriented cobordism group in dimension 5 is not trivial, see the <a href="http://en.wikipedia.org/wiki/Cobordism" rel="nofollow">Wikipedia page</a> on cobordisms.</p>