Morse homology for the Hamiltonian action in cotangent bundles

Abstract In this paper we use the gradient flow equation introduced in Asselle and Starostka (Calc. Var. 59: 113, 2020) to construct a Morse complex for the Hamiltonian action $$\mathbb {A}_H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:math> on a mixed regularity space of loops in the cotangent bundle $$T^*M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>T</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> of a closed manifold M . Connections between pairs of critical points are realized as genuine intersections between unstable and stable manifolds, which (despite being infinite dimensional objects) turn out to have finite dimensional intersection with good compactness properties. This follows from the existence of an additional structure, namely a strongly integrable (0)-essential subbundle, which behaves nicely under the negative gradient flow of the Hamiltonian action and which is needed to make comparisons. Transversality is achieved by generically perturbing the negative gradient vector field $$-\nabla \mathbb {A}_H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>∇</mml:mi> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:mrow> </mml:math> of the Hamiltonian action within a class of pseudo-gradient vector fields preserving all good compactness properties of $$-\nabla \mathbb {A}_H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>∇</mml:mi> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:mrow> </mml:math> . This follows from an abstract transversality result of independent interest for vector fields on a Hilbert manifold for which stable and unstable manifolds of rest points are infinite dimensional. The resulting Morse homology is independent of the choice of the Hamiltonian (and of all other choices but the choice of the (0)-essential subbundle, which however only changes the Morse-complex by a shift of the indices) and is isomorphic to the Floer homology of $$T^*M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>T</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> as well as to the singular homology of the free loop space of M .