Loop Space Homology Research Articles

Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups

Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups

Reflexive homology

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

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 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 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 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 using linear Hamiltonians and the square root of the energy functional.

E-infinity structure in hyperoctahedral homology

Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer-Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also give an explicit description of hyperoctahedral homology in degree zero. Combining this description and the Pontryagin product we show that hyperoctahedral homology fails to preserve Morita equivalence.

Topological coHochschild homology and the homology of free loop spaces

We study the homology of free loop spaces via techniques arising from the theory of topological coHochschild homology (coTHH). Topological coHochschild homology is a topological analogue of the classical theory of coHochschild homology for coalgebras. We produce new spectrum-level structure on coTHH of suspension spectra as well as new algebraic structure in the coBökstedt spectral sequence for computing coTHH. These new techniques allow us to compute the homology of free loop spaces in several new cases, extending known calculations.

Symplectic homology of fiberwise convex sets and homology of loop spaces

For any nonempty, compact and fiberwise convex set $K$ in $T^*\mathbb{R}^n$, we prove an isomorphism between symplectic homology of $K$ and a certain relative homology of loop spaces of $\mathbb{R}^n$. We also prove a formula which computes symplectic homology capacity (which is a symplectic capacity defined from symplectic homology) of $K$ using homology of loop spaces. As applications, we prove (i) symplectic homology capacity of any convex body is equal to its Ekeland-Hofer-Zehnder capacity, (ii) a certain subadditivity property of the Hofer-Zehnder capacity, which is a generalization of a result previously proved by Haim-Kislev.

Odd characteristic classes in entire cyclic homology and equivariant loop space homology

Given a compact manifold M and a smooth map g\colon M\to U(l\times l;\mathbb{C}) from M to the Lie group of unitary l\times l matrices with entries in \mathbb{C} , we construct a Chern character \mathrm{Ch}^-(g) which lives in the odd part of the equivariant (entire) cyclic Chen-normalized cyclic complex \mathscr{N}_{\epsilon}(\Omega_{\mathbb{T}}(M\times \mathbb{T})) of M , and which is mapped to the odd Bismut–Chern character under the equivariant Chen integral map. It is also shown that the assignment g\mapsto \mathrm{Ch}^-(g) induces a well-defined group homomorphism from the K^{-1} theory of M to the odd homology group of \mathscr{N}_{\epsilon}(\Omega_{\mathbb{T}}(M\times \mathbb{T})) .

The Milnor-Moore theorem for L_infty algebras in rational homotopy theory

We give a construction of the universal enveloping A_infty algebra of a given L_infty algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem. This proposes a new A_infty model for simply connected rational homotopy types, and uncovers a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.

Loop space homology of a small category

In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod $p$ homology of $\Omega(BG^\wedge_p)$, when $G$ is a finite group, $BG^\wedge_p$ is the $p$-completion of its classifying space, and $\Omega(BG^\wedge_p)$ is the loop space of $BG^\wedge_p$. The main purpose of this work is to shed new light on Benson's result by extending it to a more general setting. As a special case, we show that if $\mathcal{C}$ is a small category, $|\mathcal{C}|$ is the geometric realization of its nerve, $R$ is a commutative ring, and $|\mathcal{C}|^+_R$ is a "plus construction" for $|\mathcal{C}|$ in the sense of Quillen (taken with respect to $R$-homology), then $H_*(\Omega(|\mathcal{C}|^+_R);R)$ can be described as the homology of a chain complex of projective $R\mathcal{C}$-modules satisfying a certain list of algebraic conditions that determine it uniquely up to chain homotopy. Benson's theorem is now the case where $\mathcal{C}$ is the category of a finite group $G$, $R=\mathbb{F}_p$ for some prime $p$, and $|\mathcal{C}|^+_R=BG^\wedge_p$.

On the moduli space of flat symplectic surface bundles

In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. Similar to discrete surface diffeomorphisms [Nar17b], we construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology of certain infinite loop spaces. We use these infinite loop spaces to study characteristic classes of surface bundles whose holonomy groups are area preserving, in particular we give a homotopy theoretic proof of the main theorem in [KM07].

Koszul A∞-algebras and free loop space homology

AbstractWe introduce a notion of Koszul A∞-algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A∞-algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.

Homotopy groups of certain highly connected manifolds via loop space homology

For $n\geq 2$ we consider $(n-1)$-connected closed manifolds of dimension at most $(3n-2)$. We prove that away from a finite set of primes, the $p$-local homotopy groups of $M$ are determined by the dimension of the space of indecomposable elements in the cohomology ring $H^\ast(M; \mathbb{Q})$. Moreover, we show that these $p$-local homotopy groups can be expressed as a direct sum of $p$-local homotopy groups of spheres. This generalizes some of the results of our earlier work [1].

BV structure on the Hochschild cohomology of Sullivan algebras

Abstract Let X be a closed, simply connected manifold of dimension m and LX the space of free loops on X . If (∧ V, d ) is the minimal Sullivan model of X where V is finite dimensional, then there is a Gerstenhaber algebra ( ∧ V ⊗ ∧ s − 1 V # , d 0 ) , where V # is the graded dual of V , and its homology is isomorphic to the loop space homology H * ( L X ) . In this paper we define a BV structure on ( ∧ V ⊗ ∧ s − 1 V # , d 0 ) which extends the Gerstenhaber bracket.

Free loop space homology of highly connected manifolds

Abstract We calculate the homology of the free loop space of ( n - 1 ) ${(n-1)}$ -connected closed manifolds of dimension at most 3 ⁢ n - 2 ${3n-2}$ ( n ≥ 2 ${n\geq 2}$ ), with the Chas–Sullivan loop product and loop bracket. Over a field of characteristic zero, we obtain an expression for the BV-operator. We also give explicit formulas for the Betti numbers, showing they grow exponentially. Our main tool is the connection between formality, coformality and Koszul algebras that was elucidated by the first author [6].

The free loop space homology of (n-1)-connected 2n-manifolds

Our goal in this paper is to compute the integral free loop space homology of (n-1)-connected 2n-manifolds. We do this when nge 2 and nne 2,4,8, though the techniques here should cover a much wider range of manifolds. We also give partial information concerning the action of the Batalin–Vilkovisky operator.

Hochschild Cohomology of a Sullivan Algebra

Let \({A= (\land V, d)}\) be a minimal Sullivan algebra where V is finite dimensional. We show that the Hochschild cohomology HH*(A; A) can be computed in terms of derivations of A. This provides another method to compute the loop space homology of a simply connected space for which \({ \pi_*(X) \otimes \mathbb{Q} }\) is finite dimensional.

Derived string topology and the Eilenberg-Moore spectral sequence

Let M be a simply-connected closed manifold of dimension m. Chas and Sullivan have defined (co)products on the homology of the free loop space H*(LM). Felix and Thomas have extended the loop (co)products to those of simply-connected Gorenstein spaces over a field. We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. In Algebraic Topology, one of the most important tools for computing the (co)homology of the space of free loops on a space is the (co)homological Eilenberg-Moore spectral sequence. Consider, over any field, the homological Eilenberg-Moore spectral sequence converging to H *(LM). Our description of the loop product enables one to conclude that this spectral sequence is multiplicative with respect to the Chas-Sullivan loop product and that its E 2-term is the Hochschild cohomology of H*(M). This gives a new method to compute the loop products on H *(LS m ) and H *(LℂP r ), the free loop space homology of spheres and complex projective spaces.

Power Operations and Coactions in Highly Commutative Homology Theories

Power operations in the homology of infinite loop spaces, and H_\infty or E_\infty ring spectra have a long history in Algebraic Topology. In the case of ordinary mod p homology for a prime p , the power operations of Kudo, Araki, Dyer and Lashof interact with Steenrod operations via the Nishida relations, but for many purposes this leads to complicated calculations once iterated applications of these functions are required. On the other hand, the homology coaction turns out to provide tractable formulae better suited to exploiting multiplicative structure. We show how to derive suitable formulae for the interaction between power operations and homology coactions in a wide class of examples; our approach makes crucial use of modern frameworks for spectra with well behaved smash products. In the case of mod p homology, our formulae extend those of Bisson and Joyal to odd primes. We also show how to exploit our results in sample calculations, and produce some apparently new formulae for the Dyer-Lashof action on the dual Steenrod algebra.

Filtered Hopf algebras and counting geodesic chords

We prove lower bounds on the growth of certain filtered Hopf algebras by means of a Poincare–Birkhoff–Witt type theorem for ordered products of primitive elements. When applied to the loop space homology algebra endowed with a natural length-filtration, these bounds lead to lower bounds for the number of geodesic paths between two points. Specifically, given a closed manifold \(M\) whose universal covering space is not homotopy equivalent to a finite complex and whose fundamental group has polynomial growth, for any Riemannian metric on \(M\), any pair of non-conjugate points \(p,q \in M\), and every component \({\mathcal C}\) of the space of paths from \(p\) to \(q\), the number of geodesics in \({\mathcal C}\) of length at most \(T\) grows at least like \(e^{\sqrt{T}}\). Using Floer homology, we extend this lower bound to Reeb chords on the spherisation of \(M\), and give a lower bound for the volume growth of the Reeb flow.

Corrigendum: On the Floer Homology of Cotangent Bundles

We fix an orientation issue that appears in our previous paper about the isomorphism between Floer homology of cotangent bundles and loop space homology. When the second Stiefel‐Whitney class of the underlying manifold does not vanish on 2‐tori, this isomorphism requires the use of a twisted version of the Floer complex. © 2014 Wiley Periodicals, Inc.