Morse theory, closed geodesics, and the homology of free loop spaces

This is a survey paper on Morse theory and the existence problem for closed geodesics. The free loop space plays a central role, since closed geodesics are critical points of the energy functional. As such, they can be analyzed through variational methods. The topics that we discuss include: Riemannian background, the Lyusternik-Fet theorem, the Lyusternik-Schnirelmann principle of subordinated classes, the Gromoll-Meyer theorem, Bott's iteration of the index formulas, homological computations using Morse theory, $SO(2)$- vs. $O(2)$-symmetries, Katok's examples and Finsler metrics, relations to symplectic geometry, and a guide to the literature. The Appendix written by Umberto Hryniewicz gives an account of the problem of the existence of infinitely many closed geodesics on the $2$-sphere.Comment: 45 pages, 5 figures. Appendix by Umberto Hryniewic

Symplectic homology and the Eilenberg-Steenrod axioms

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer-Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology with a secondary coproduct on positive symplectic homology. In the appendix, joint with Peter Albers, we discuss obstructions to the existence of certain Liouville cobordisms.Comment: v3: corrected Lemma 7.11. Various other minor modifications and reformatting. Final version to be published in Algebraic and Geometric Topolog

Fredholm theory and transversality for the parametrized and for the S1S^1-invariant symplectic action

We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the $L^2$-gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic $S^1$-invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define $S^1$-equivariant Floer homology. As an intermediate result of independent interest, we generalize Aronszajn's unique continuation theorem to a class of elliptic integro-differential inequalities of order two.Comment: 63 page

Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces

We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between linearized contact homology of a fillable contact manifold and symplectic homology of its filling.Comment: Final version, 92 pages, 7 figures, index of notations. Remark 4.9 in Version 1 is wrong. To correct it we modified the weights for the Sobolev spaces along the gradient trajectories (Figure 3). Proposition 4.8 in Version 1 is now split into Propositions 4.8 and 4.9. This version contains full details for the second derivative estimates in Lemma 4.2

The space of paths in complex projective space with real boundary conditions

We compute the integral homology of the space of paths in $\mathbb{C}P^n$ with endpoints in $\mathbb{R}P^n$, $n \ge 1$ and its algebra structure with respect to the Pontryagin-Chas-Sullivan product with $\mathbb{Z}/2$-coefficients.Comment: 33 pages, 6 figure

Posets of annular non-crossing partitions of types B and D

We study the set \sncb (p,q) of annular non-crossing permutations of type B, and we introduce a corresponding set \ncb (p,q) of annular non-crossing partitions of type B, where $p$ and $q$ are two positive integers. We prove that the natural bijection between \sncb (p,q) and \ncb (p,q) is a poset isomorphism, where the partial order on \sncb (p,q) is induced from the hyperoctahedral group $B_{p+q}$, while \ncb (p,q) is partially ordered by reverse refinement. In the case when $q=1$, we prove that \ncb (p,1) is a lattice with respect to reverse refinement order. We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between \sncd (p,q) and \ncd (p,q). For $q=1$, the poset \ncd (p,1) coincides with a poset ``$NC^{(D)} (p+1)$'' constructed in a paper by Athanasiadis and Reiner in 2004, and is a lattice by the results of that paper.Comment: Revised version (shortened Introduction, corrected typos), 31 pages, 4 figures, to appear in Discrete Mathematic