arXiv subjects

Alexandru Oancea

Publications and source records attributed to Alexandru Oancea.

At least 19 recordsLinked to original sources

arXiv2026

Resonances and string point invertibility for compact rank one symmetric spaces

We calculate the Batalin-Vilkovisky (BV) algebra structure of Rabinowitz loop homology for compact rank one symmetric spaces. As a consequence, we prove that such a space satisfies a natural homological condition called string point invertibility if and only if its Euler characteristic is equal to the characteristic of the coefficient field for loop homology. This implies certain cases of Viterbo's conjecture on a uniform bound on the spectral norm of exact Lagrangian submanifolds in cotangent disk bundles. Furthermore, we prove that whenever a compact rank one symmetric space is string point invertible, the critical levels of its loop homology classes with respect to an arbitrary Riemannian metric satisfy a resonance condition with respect to degrees and a density condition for closed geodesics. This generalizes results of Hingston and Rademacher for spheres to a broader class of compact rank one symmetric spaces.

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arXiv2026

Open problems in billiards and quantitative symplectic geometry

This document collects contributions to the Open Problem List in Billiards and Quantitative Symplectic Geometry, compiled following discussions during the workshop ``Billiards and quantitative symplectic geometry'' that took place at the University of Heidelberg on July 14--18, 2025.

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arXiv2024

Morse homology with DG coefficients

We develop a theory of Morse homology and cohomology with coefficients in a derived local system, for manifolds and also more generally for colimits of spaces that have the homotopy type of manifolds, with a view towards Floer theory. The model that we adopt for derived, or differential graded (DG) local systems is that of DG modules over chains on the based loop space of a manifold. These encompass both classical (non DG) local systems and chains on fibers of Hurewicz fibrations. We prove that the Morse homology and cohomology groups that we construct are isomorphic to DG Tor and Ext functors. The key ingredient in the definition is a notion of twisting cocycle obtained by evaluating into based loops a coherent system of representatives for the fundamental classes of the moduli spaces of Morse trajectories of arbitrary dimensions. From this perspective, our construction sits midway between classical Morse homology with twisted coefficients and more refined invariants of Floer homotopical flavor. The construction of the twisting cocycle is originally due to Barraud and Cornea with Z/2-coefficients. We show that the twisting cocycle with integer coefficients is equivalent to Brown's universal twisting cocycle. We prove that Morse homology with coefficients in chains on the fiber of a Hurewicz fibration recovers the homology of the total space of the fibration. We study several structural properties of the theory: invariance, functoriality, and Poincaré duality, also in the nonorientable case.

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arXiv2024

Topological Frobenius algebras

We define the notions of unital/counital/biunital infinitesimal anti-symmetric bialgebras and coFrobenius bialgebras and discuss their algebraic properties. We also define the notion of a graded 2D open-closed TQFT. These structures arise in Rabinowitz Floer homology, loop space homology, quantum homology, and the homology of finite dimensional manifolds. The underlying vector spaces, which are typically infinite dimensional, belong to a class of topological vector spaces known as Tate vector spaces.

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arXiv2024

Rabinowitz Floer homology as a Tate vector space

We show that the category of linearly topologized vector spaces over discrete fields constitutes the correct framework for algebraic structures on Floer homologies with field coefficients. Our case in point is the Poincaré duality theorem for Rabinowitz Floer homology. We prove that Rabinowitz Floer homology is a locally linearly compact vector space in the sense of Lefschetz, or, equivalently, a Tate vector space in the sense of Beilinson-Feigin-Mazur. Poincaré duality and the graded Frobenius algebra structure on Rabinowitz Floer homology then hold in the topological sense. Along the way, we develop in a largely self-contained manner the theory of linearly topologized vector spaces, with special emphasis on duality and completed tensor products, complementing results of Beilinson-Drinfeld, Beilinson, Rojas, Positselski, and Esposito-Penkov.

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arXiv2024

Floer Homology with DG Coefficients. Applications to cotangent bundles

We define Hamiltonian Floer homology with differential graded (DG) local coefficients for symplectically aspherical manifolds. The differential of the underlying complex involves chain representatives of the fundamental classes of the moduli spaces of Floer trajectories of arbitrary dimension. This setup allows in particular to define and compute Floer homology with coefficients in chains on fibers of fibrations over the free loop space of the underlying symplectic manifold. We develop the DG Floer toolset, including continuation maps and homotopies, and we also define and study symplectic homology groups with DG local coefficients. We define spectral invariants and establish general criteria for almost existence of contractible periodic orbits on regular energy levels of Hamiltonian systems inside Liouville domains. In the case of cotangent bundles, we prove a Viterbo isomorphism theorem with DG local coefficients. This serves as a stepping stone for applications to the almost existence of contractible closed characteristics on closed smooth hypersurfaces. In this context, our methods allow to access for the first time the dichotomy between closed manifolds that are aspherical and those that are not.

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arXiv2024

BV bialgebra structures in Floer theory and string topology

We derive the notions of BV unital infinitesimal bialgebra and BV Frobenius algebra from the topology of suitable compactifications of moduli spaces of decorated genus 0 curves. We construct these structures respectively on reduced symplectic homology and Rabinowitz Floer homology. As an application, we construct these structures in nonequivariant string topology. We also show how the Lie bialgebra structure in equivariant string topology, and more generally on $S^1$-equivariant symplectic homology, is obtained as a formal consequence.

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arXiv2024

Multiplicative structures on cones and duality

We initiate the study of multiplicative structures on cones and show that cones of Floer continuation maps fit naturally in this framework. We apply this to give a new description of the multiplicative structure on Rabinowitz Floer homology and cohomology, and to give a new proof of the Poincaré duality theorem which relates the two. The underlying algebraic structure admits two incarnations, both new, which we study and compare: on the one hand the structure of $A_2^+$-algebra on the space $\mathcal{A}$ of Floer chains, and on the other hand the structure of $A_2$-algebra involving $\mathcal{A}$, its dual $\mathcal{A}^\vee$ and a continuation map from $\mathcal{A}^\vee$ to $\mathcal{A}$.

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arXiv2023

Loop coproduct in Morse and Floer homology

By a well-known theorem of Viterbo, the symplectic homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. In this paper we extend the scope of this isomorphism in several directions. First, we give a direct definition of {\em Rabinowitz loop homology} in terms of Morse theory on the loop space and prove that its product agrees with the pair-of-pants product on Rabinowitz Floer homology. The proof uses compactified moduli spaces of punctured annuli. Second, we prove that, when restricted to {\em positive} Floer homology, resp.~loop space homology relative to the constant loops, the Viterbo isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. Third, we introduce {\em reduced loop homology}, which is a common domain of definition for a canonical reduction of the loop product and for extensions of the loop homology coproduct which together define the structure of a commutative cocommutative unital infinitesimal anti-symmetric bialgebra. Along the way, we show that the Abbondandolo-Schwarz quasi-isomorphism going from the Floer complex of quadratic Hamiltonians to the Morse complex of the energy functional can be turned into a filtered chain isomorphism by using linear Hamiltonians and the square root of the energy functional.

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arXiv2022

Reduced symplectic homology and string topology

We introduce a common domain of definition for the loop product and the loop coproduct, reduced loop homology, on which they combine to a unital infinitesimal anti-symmetric bialgebra structure. In particular, a relation conjectured by Sullivan holds with an extra term. The structure depends on choices governed by secondary continuation maps. These results on string topology are proved in the more general context of reduced symplectic homology for a suitable class of Weinstein manifolds.

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arXiv2020

Poincar\'e duality for loop spaces

We show that Rabinowitz Floer homology and cohomology carry the structure of a graded Frobenius algebra for both closed and open strings. We prove a Poincar\'e duality theorem between homology and cohomology that preserves this structure. This lifts to a duality theorem between graded open-closed TQFTs. We use in a systematic way the formalism of Tate vector spaces. Specializing to the case of cotangent bundles, we define Rabinowitz loop homology and cohomology and explain from a unified perspective pairs of dual results that have been observed over the years in the context of the search for closed geodesics. These concern critical levels, relations to the based loop space, manifolds all of whose geodesics are closed, Bott index iteration, and level-potency. Moreover, the graded Frobenius algebra structure gives meaning and proof to a relation conjectured by Sullivan between the loop product and coproduct.

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arXiv2020

The Deligne-Mumford operad as a trivialization of the circle action

We prove that the tree-like Deligne-Mumford operad is a homotopical model for the trivialization of the circle in the higher-genus framed little discs operad. Our proof is based on a geometric argument involving nodal annuli. We use as a model for the higher-genus framed little discs an operad of Riemann surfaces with analytically parametrized boundary. We develop the formalism of topological moduli problems as a framework to accommodate the orbifold nature of the Deligne-Mumford operad.

math.AT↗

arXiv2018

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.

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arXiv2016

Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity

In this article we examine under which conditions symplectic homology with local coefficients of a unit disk bundle $D^*M$ vanishes. For instance this is the case if the Hurewicz map $π_2(M)\to H_2(M;\mathbb{Z})$ is nonzero. As an application we prove finiteness of the $π_1$-sensitive Hofer-Zehnder capacity of unit disk bundles in these cases. We also prove uniruledness for such cotangent bundles. Moreover, we find an obstruction to the existence of $H$-space structures on general topological spaces, formulated in terms of local systems.

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arXiv2015

Symplectic and contact differential graded algebras

We define Hamiltonian simplex differential graded algebras (DGA) with differentials that deform the high energy symplectic homology differential and wrapped Floer homology differential in the cases of closed and open strings in a Weinstein manifold, respectively. The order $m$ term in the differential is induced by varying natural degree $m$ co-products over an $(m-1)$-simplex, where the operations near the boundary of the simplex are trivial. We show that the Hamiltonian simplex DGA is quasi-isomorphic to the (non-equivariant) contact homology algebra and to the Legendrian homology algebra of the ideal boundary in the closed and open string cases, respectively.

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arXiv2014

$S^1$-equivariant symplectic homology and linearized contact homology

We present three equivalent definitions of $S^1$-equivariant symplectic homology. We show that, using rational coefficients, the positive part of $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, and introduce a rigorously defined substitute for cylindrical/linearized contact homology based on an $S^1$-equivariant construction.

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arXiv2014

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.

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arXiv2013

The Gysin exact sequence for $S^1$-equivariant symplectic homology

We define $S^1$-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space.

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