Loop Space Homology Research Articles

Regularity in the growth of the loop space homology of a finite CW complex

To any path connected topological space X X , such that rk ⁡ H i ( X ) > ∞ \operatorname {rk} H_i(X) >\infty for all i ≥ 0 i\geq 0 , are associated the following two sequences of integers: b i = rk ⁡ H i ( Ω X ) b_i= \operatorname {rk} H_i(\Omega X) and r i = rk ⁡ π i + 1 ( X ) r_i= \operatorname {rk} \pi _{i+1}(X) . If X X is simply connected, the Milnor-Moore theorem together with the Poincaré-Birkoff-Witt theorem provides an explicit relation between these two sequences. If we assume moreover that H i ( X ; Q ) = 0 H_i(X;\mathbb Q)=0 , for all i ≫ 0 i\gg 0 , it is a classical result that the sequence of Betti numbers ( b i ) (b_i) grows polynomially or exponentially, depending on whether the sequence ( r i ) (r_i) is eventually zero or not. The purpose of this note is to prove, in both cases, that the r t h r^{\mathrm {th}} Betti number b r b_r is controlled by the immediately preceding ones. The proof of this result is based on a careful analysis of the Sullivan model of the free loop space X S 1 X^{S^1} .

Hochschild cohomology of a strongly homotopy commutative algebra

The Hochschild cohomology of a DG algebra A with coefficients in itself is, up to a suspension of degrees, a graded Lie algebra. The purpose of this paper is to prove that a certain DG Lie algebra of derivations appears as a finite codimensional graded sub Lie algebra of this Lie algebra when A is a strongly homotopy commutative algebra whose homology is concentrated in finitely many degrees. This result has interesting implications for the free the loop space homology which we explore here as well.

The Growth Rate of Symplectic Homology and Affine Varieties

We will show that the cotangent bundle of a manifold whose free loopspace homology grows exponentially is not symplectomorphic to any smooth affine variety. We will also show that the unit cotangent bundle of such a manifold is not Stein fillable by a Stein domain whose completion is symplectomorphic to a smooth affine variety. For instance, these results hold for end connect sums of simply connected manifolds whose cohomology with coefficients in some field has at least two generators. We use an invariant called the growth rate of symplectic homology to prove this result.

Floer homology for magnetic fields with at most linear growth on the universal cover

The Floer homology of a cotangent bundle is isomorphic to loop space homology of the underlying manifold, as proved by Abbondandolo and Schwarz, Salamon and Weber, and Viterbo. In this paper we show that in the presence of a Dirac magnetic monopole which admits a primitive with at most linear growth on the universal cover, the Floer homology in atoroidal free homotopy classes is again isomorphic to loop space homology. As a consequence we prove that for any atoroidal free homotopy class and any sufficiently small τ>0, any magnetic flow associated to the Dirac magnetic monopole has a closed orbit of period τ belonging to the given free homotopy class. In the case where the Dirac magnetic monopole admits a bounded primitive on the universal cover we also prove the Conley conjecture for Hamiltonians that are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits.

Loop space homology associated with the mod 2 Dickson invariants

The spaces BG2 and BDI.4/ have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces, one has the classifying spaces of the finite groups BG2.q/, for an odd prime powerq, and the exotic family of classifying spaces of 2-local finite groupsBSol.q/. In this article we compute the loop space homology of BG2.q/ ^ andBSol.q/ for all odd primesq, as Hopf algebras over the Steenrod algebra, and the associated Bockstein spectral sequences.

Spectral sequences in string topology

In this paper, we investigate the behaviour of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specificially with the Chas-Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fibre bundles: the fibre bundle $\Omega^n M \to L^n M \to M$ for an $h_*$-oriented manifold $M$ and the looped fibre bundle $L^n F \to L^n E \to L^n B$ of a fibre bunde $F \to E \to B$ of $h_*$-oriented manifolds. Our method lies in the construction of Gysin morphisms of spectral sequences. We apply these results to study the ordinary homology of the free loop spaces of sphere bundles and generalized homologies of the free loop spaces of spheres and projective spaces.

Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations

AbstractWill J. Merry computed Rabinowitz Floer homology above Mañé's critical value in terms of loop space homology in [14] by establishing an Abbondandolo–Schwarz short exact sequence. The purpose of this paper is to provide an alternative proof of Merry's result. We construct a continuation homomorphism for symplectic deformations which enables us to reduce the computation to the untwisted case. Our construction takes advantage of a special version of the isoperimetric inequality which above Mañé's critical value holds true.

On the mapping space homotopy groups and the free loop space homology groups

On the mapping space homotopy groups and the free loop space homology groups

Coalgebraic tensor product and homology operations

In their paper [4] Hunton and Turner set up the framework for homological algebra of coalgebraic rings and coalgebraic modules, which arise naturally in the study of ordinary and generalized homologies of infinite loop spaces. Their language has proved to be a useful tool both for describing (see eg Hunton and Turner [5] and Kashiwabara and Wilson [7]) and computing (see Kashiwabara [6]) homology of infinite loop spaces. Of course, such objects tend to get equipped with extra structures. For example the Steenrod algebra [11], the Dyer–Lashof algebra [3] or the algebra of cohomology operations for appropriate generalized cohomology theory (see Adams [1]) may act on these objects. Thus it is natural to ask if these actions can be extended to the algebraic constructions defined by Hunton and Turner, notably to the coalgebraic tensor product.

THE INTEGRAL PONTRJAGIN HOMOLOGY OF THE BASED LOOP SPACE ON A FLAG MANIFOLD

The based loop space homology of a special family of homogeneous spaces, flag manifolds of connected compact Lie groups is studied. First, the rational homology of the based loop space on a complete flag manifold is calculat ed together with its Pontrjagin structure. Second, it is shown that the integral homology of the based loop space on a flag manifold is torsion free. This results in a description of the integral homology. In addition, the integral Pontrjagin st ructure is determined.

Graph products of spheres, associative graded algebras and Hilbert series

Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Non-commutative Grobner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment-angle complex. We apply our result to the loop-space homology of this space.

Correction to “Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology”

Manfred Stelzer has pointed out that part of Corollary 4.5 of our paper "Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology" [Alg. Geo. Top. 8 (2008), 2109--2129] was not sufficiently proved, and, indeed, is likely incorrect as stated. This necessitates a little more argument to finish the proof of the main theorem of the original paper. The statement of this theorem, and all the examples, remain unchanged.

Van den Bergh isomorphisms in string topology

Let M be a path-connected closed oriented d -dimensional smooth manifold and let \mathbb{k} be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of M , H_{*+d}(\mathrm{L}\mkern-1.5mu M ) is a Batalin–Vilkovisky algebra. Let G be a topological group such that M is a classifying space of G . Denote by S_*(G) the (normalized) singular chains on G . Suppose that G is discrete or path-connected. We show that there is a Van Den Bergh type isomorphism \mathrm{HH}^{-p}(S_*(G),S_*(G))\cong \mathrm{HH}_{p+d}(S_*(G),S_*(G)). Therefore, the Gerstenhaber algebra \mathrm{HH}^{*}(S_*(G),S_*(G)) is a Batalin–Vilkovisky algebra and we have a linear isomorphism \mathrm{HH}^{*}(S_*(G),S_*(G))\cong H_{*+d}(\mathrm{L}\mkern-1.5mu M ). This linear isomorphism is expected to be an isomorphism of Batalin–Vilkovisky algebras. We also give a new characterization of Batalin–Vilkovisky algebra in terms of the derived bracket.

Hodge-type decomposition in the homology of long knots

The paper describes a natural splitting in the rational homology and homotopy of the spaces of long knots. This decomposition presumably arises from the cabling maps in the same way as a natural decomposition in the homology of loop spaces arises from power maps. The generating function for the Euler characteristics of the terms of this splitting is presented. Based on this generating function, it is shown that both the homology and homotopy ranks of the spaces in question grow at least exponentially. Using natural graph-complexes, one shows that this splitting on the level of the bialgebra of chord diagrams is exactly the splitting defined earlier by Dr. Bar-Natan. The appendix presents tables of computer calculations of the Euler characteristics. These computations give a certain optimism that the Vassiliev invariants of order greater than 20 can distinguish knots from their inverses.

Higher commutators in the loop space homology of K-products

We consider a problem of calculating the loop space homology for so-called polyhedral products defined by an arbitrary simplicial complex K. A presentation of this homology algebra is obtained from the homology of the complements of diagonal subspace arrangements, which, in turn, is calculated using an infinite resolution of the exterior Stanley-Reisner algebra. We get an explicit presentation of the loop homology algebra for polyhedral products for classes of simplicial complexes such as flag complexes and the duals of sequentially Cohen-Macaulay complexes in terms of higher commutator products. We give a construction for the iteration of higher products and discuss the relationship between this problem and problems in commutative algebra.

Loop coproducts in string topology and triviality of higher genus TQFT operations

Cohen and Godin constructed a positive boundary topological quantum field theory (TQFT) structure on the homology of free loop spaces of oriented closed smooth manifolds by associating certain operations called string operations to orientable surfaces with parametrized boundaries. We show that all TQFT string operations associated to surfaces of genus at least one vanish identically. This is a simple consequence of properties of the loop coproduct which will be discussed in detail. One interesting property is that the loop coproduct is nontrivial only on the degree d homology group of the connected component of L M consisting of contractible loops, where d = dim M , with values in the degree 0 homology group of constant loops. Thus the loop coproduct behaves in a dramatically simpler way than the loop product.

Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology

We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence.

Structure relations in special A ∞-bialgebras

We compute the structure relations in special A ∞-bialgebras whose operations are limited to those defining the underlying A ∞-(co)algebra substructure. Such bialgebras appear as the homology of certain loop spaces. Whereas structure relations in general A ∞-bialgebras depend upon the combinatorics of permutahedra, only Stasheff’s associahedra are required here.

An algebraic model for the loop space homology of a homotopy fiber

Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF, the simplicial (Kan) loop group on F. To construct this model, we develop machinery for modeling the homotopy fiber of a morphism of chain Hopf algebras. Essential to our construction is a generalization of the operadic description of the category DCSH of chain coalgebras and of strongly homotopy coalgebra maps given in (math.AT/0505559) to strongly homotopy morphisms of comodules over Hopf algebras. This operadic description is expressed in terms of a general theory of monoidal structures in categories with morphism sets parametrized by co-rings, which we elaborate here.

Dyer–Lashof–Cohen operations in Hochschild cohomology

We give explicit formulae for operations in Hochschild cohomology which are analogous to the operations in the homology of double loop spaces. As a corollary we obtain that any brace algebra in finite characteristic is always a restricted Lie algebra.