User stephan mescher - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:24:24Z http://mathoverflow.net/feeds/user/13326 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112581/induced-maps-in-morse-homology/115883#115883 Answer by Stephan Mescher for Induced maps in Morse Homology Stephan Mescher 2012-12-09T12:26:37Z 2012-12-09T12:26:37Z <p>A direct description of functoriality in Morse homology is given by Abbondandolo and Schwarz in Appendix A.2 of <a href="http://arxiv.org/abs/0810.1995" rel="nofollow">"Floer homology of cotangent bundles and the loop product"</a>.</p> <p>If $\varphi: M_1 \to M_2$ is a differentiable map between manifolds, $x$ is a critical point of the Morse function on $M_1$ and $y$ is a critical point of the Morse function on $M_2$, the chain map induced by $\varphi$ is then roughly given by making the intersections $$\varphi(W^u(x))\cap W^s(y)$$ transverse and counting elements of zero-dimensional intersections. Transversality can easily be achieved by perturbing the Riemannian metrics on $M_1$ and $M_2$.</p> http://mathoverflow.net/questions/55129/transversality-in-morse-theory-for-the-perturbed-geodesic-action-functional/57003#57003 Answer by Stephan Mescher for Transversality in Morse theory for the (perturbed) geodesic action functional Stephan Mescher 2011-03-01T14:00:11Z 2011-03-01T14:00:11Z <p>Instead of considering the Morse homology of the energy functional my making it Morse-Smale, it might be easier for you (and geometrically more natural) to view the geodesic energy as a Morse-Bott functional, whose critical points appear in $S^1$-families. </p> <p>The appropriate setting of Morse homology for Morse-Bott functions was worked out by Urs Frauenfelder in the Appendix of "The Arnold-Givental conjecture and moment Floer homology".</p> <p>The Morse(-Bott) homology of the geodesic functional was considered by Abbondandolo and Schwarz in section 4 of "Estimates and Computations in Rabinowitz-Floer homology", see here:</p> <p><a href="http://arxiv.org/abs/0907.1976" rel="nofollow">Abbdondandolo-Schwarz</a></p>