# Transversality for local Morse homology with symmetries and applications

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## Abstract

We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive perturbation process indexed by the strata of the isotropy set. A global existence theorem for symmetric Morse–Smale pairs is also proved. Regarding applications, we focus on Hamiltonian dynamics and rigorously establish a local contact homology package based on discrete action functionals. We prove a persistence theorem, analogous to the classical shifting lemma for geodesics, asserting that the iteration map is an isomorphism for good and admissible iterations. We also consider a Chas–Sullivan product on non-invariant local Morse homology, which plays the role of pair-of-pants product, and study its relationship to symplectically degenerate maxima. Finally, we explore how our invariants can be used to study bifurcation of critical points (and periodic points) under additional symmetries.

## Notes

### Acknowledgements

We are grateful to Viktor Ginzburg for useful comments regarding this paper. UH and LM would like to thank J. Fish, M. Hutchings, J. Nelson and K. Wehrheim for organizing the AIM Workshop “Transversality in contact homology” in December 2014, where Morse homology in the presence of symmetries was a topic of intense discussion. UH is extremely grateful to Alberto Abbondandolo for numerous insightful conversations concerning the analysis involved in this work. UH also thanks the Floer Center of Geometry (Bochum) for its warm hospitality, and acknowledges the support from CNPq (Brazil) and also the generous support of the Alexander von Humboldt Foundation during the preparation of this manuscript. LM was partially supported by CNPq (Brazil).

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