Quasi-transitive maps are the homogeneous maps on the plane with finite orbits under the action of their automorphism groups. We show that there exist quasi-transitive maps of the types [p 3 , 3] for p odd and p ≥ 5, but there doesn't exist vertex-transitive map of such types. In particular, we determine the surface with the lowest possible genus that admit a polyhedral map of the type [5 3 , 3].Theorem 1.1. There exist quasi-transitive maps on the plane of the types [p 3 , 3], p odd, p ≥ 5.
In this paper, we show that the Chas-Sullivan product (respectively the Goresky-Hingston product) on level homology is modeled on local geometry of an isolated closed geodesic with slowest (resp. fastest) possible index growth rate. We discuss how string topology along with this result gives a new perspective on questions of the existence of closed geodesics.
The Goresky-Hingston coproduct was first introduced by D. Sullivan and later extended by M. Goresky and N. Hingston. In this article we give a Morse theoretic description of the coproduct. Using the description we prove homotopy invariance property of the coproduct. We describe a connection between our Morse theoretic coproduct and a coproduct on Floer homology of cotangent bundle.
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