Double Loop Space Research Articles

The (anti-)holomorphic sector in [formula omitted]-equivariant cohomology, and the Witten class

Atiyah's classical work on circular symmetry and stationary phase shows how the Aˆ-genus is obtained by formally applying the equivariant cohomology localization formula to the loop space of a simply connected spin manifold. The same technique, applied to a suitable “antiholomorphic sector” in the C/Λ-equivariant cohomology of the conformal double loop space Maps(C/Λ,X) of a rationally string manifold X produces the Witten genus of X. This can be seen as an equivariant localization counterpart to Berwick-Evans supersymmetric localization derivation of the Witten genus.

Bigerbes

The bigerbes introduced here give a refinement of the notion of 2-gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form 'bundle 2-gerbes' in two ways; this structure replaces higher associativity conditions. We provide natural examples, including a Brylinski-McLaughlin bigerbe associated to a principal G-bundle for a simply connected simple Lie group. This represents the first Pontryagin class of the bundle, and is the obstruction to the lifting problem on the associated principal bundle over the loop space to the structure group consisting of a central extension of the loop group; in particular, trivializations of this bigerbe for a spin manifold are in bijection with string structures on the original manifold. Other natural examples represent 'decomposable' 4-classes arising as cup products, a universal bigerbe on K(Z,4) involving its based double loop space, and the representation of any 4-class on a space by a bigerbe involving its free double loop space. The generalization to 'multigerbes' of arbitrary degree is also described.

A construction of complex analytic elliptic cohomology from double free loop spaces

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\mathcal {C}$, the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$.

The fibre of the degree 3 map, Anick spaces and the double suspension

AbstractLet S2n+1{p} denote the homotopy fibre of the degree p self map of S2n+1. For primes p ≥ 5, work by Selick shows that S2n+1{p} admits a non-trivial loop space decomposition if and only if n = 1 or p. Indecomposability in all but these dimensions was obtained by showing that a non-trivial decomposition of ΩS2n+1{p} implies the existence of a p-primary Kervaire invariant one element of order p in $\pi _{2n(p-1)-2}^S$. We prove the converse of this last implication and observe that the homotopy decomposition problem for ΩS2n+1{p} is equivalent to the strong p-primary Kervaire invariant problem for all odd primes. For p = 3, we use the 3-primary Kervaire invariant element θ3 to give a new decomposition of ΩS55{3} analogous to Selick's decomposition of ΩS2p+1{p} and as an application prove two new cases of a long-standing conjecture stating that the fibre of the double suspension $S^{2n-1} \longrightarrow \Omega ^2S^{2n+1}$ is homotopy equivalent to the double loop space of Anick's space.

Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework which is presented step-by-step with examples throughout. In this second part of two papers, we give the general categorical formulation.

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework, which is presented step-by-step with examples throughout. In this first part of two papers, we concentrate on the simplicial and operadic aspects.

Order of the canonical vector bundle over configuration spaces of spheres

Abstract Given a vector bundle, its (stable) order is the smallest positive integer t such that the t-fold self-Whitney sum is (stably) trivial. So far, the order and the stable order of the canonical vector bundle over configuration spaces of Euclidean spaces have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54], [F. R. Cohen, M. E. Mahowald and R. J. Milgram, The stable decomposition for the double loop space of a sphere, Algebraic and Geometric Topology (Stanford 1976), Proc. Sympos. Pure Math. 32 Part 2, American Mathematical Society, Providence 1978, 225–228], and [S.-W. Yang, Order of the Canonical Vector Bundle on C n ⁢ ( k ) / Σ k {C_{n}(k)/\Sigma_{k}} , ProQuest LLC, Ann Arbor, 1978]. Moreover, the order and the stable order of the canonical vector bundle over configuration spaces of closed orientable Riemann surfaces with genus greater than or equal to one have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54]. In this paper, we mainly study the order and the stable order of the canonical vector bundle over configuration spaces of spheres and disjoint unions of spheres.

Spherical classes in some finite loop spaces of spheres

We completely determine spherical classes in single, double and triple loop spaces of spheres. We also show that for l⩾4 and n>1, there exists no spherical class in H⁎ΩlSl+n in dimensions more than 2ln+2l−1(l−2)+2. This bound may not be optimal, but reduces the computation of spherical classes in H⁎ΩlSl+n to verifying finite number of cases. We also provide a table for possible spherical classes in H⁎ΩlSl+n with 3<l<9. Our results, provide positive evidence for the truth of Eccles conjecture.

Permutohedral structures on E2E_{2}-operads

Abstract There are two interesting families of E 2 {E_{2}} -operads, those that detect double loop spaces, and those that solve Deligne’s conjecture on Hochschild cochains. The first family deformation retracts to Milgram’s model obtained by gluing together permutohedra along their faces. We show how the second family can be covered by permutohedra as well, shedding new light on several proposed solutions of Deligne’s conjecture. In particular, our approach induces an explicit homotopy equivalence between the models of the two families. The permutohedra and partial orders play a central role providing direct links to other fields of mathematics. We for instance find a new cellular decomposition of permutohedra using partial orders and that the permutohedra give the cells for the Dyer–Lashof operations.

On the metastable homotopy of mod 2 Moore spaces

In this article, we study the exponents of metastable homotopy of mod $2$ Moore spaces. Our result gives that the double loop space of $4n$-dimensional mod $2$ Moore spaces has a multiplicative exponent $4$ below the range of $4$ times the connectivity. As a consequence, the homotopy groups of $4n$-dimensional mod $2$ Moore spaces have an exponent $4$ below the range of $4$ times the connectivity.

Swiss-cheese action on the totalization of action-operads

We prove that if a pair of semi-cosimplicial spaces (X,Y) arise from a coloured operad then the semi-totalization sTot(Y) has the homotopy type of a relative double loop space and the pair (sTot(X),sTot(Y)) is weakly equivalent to an explicit algebra over the two dimensional Swiss-cheese operad.

Braided injections and double loop spaces

We consider a framework for representing double loop spaces (and more generally E-2 spaces) as commutative monoids. There are analogous commutative rectifications of braided monoidal structures and we use this framework to define iterated double deloopings. We also consider commutative rectifications of E-infinity spaces and symmetric monoidal categories and we relate this to the category of symmetric spectra.

Reciprocity Laws on Algebraic Surfaces via Iterated Integrals

AbstractIn this paper we introduce new local symbols, which we call 4-function local symbols. We formulate reciprocity laws for them. These reciprocity laws are proven using a new method - multidimensional iterated integrals. Besides providing reciprocity laws for the new 4-function local symbols, the same method works for proving reciprocity laws for the Parshin symbol. Both the new 4-function local symbols and the Parshin symbol can be expressed as a finite product of newly defined bi-local symbols, each of which satisfies a reciprocity law. TheK-theoretic variant of the first 4-function local symbol is defined in the Appendix. It differs by a sign from the one defined via iterated integrals. Both the sign and theK-theoretic variant of the 4-function local symbol satisfy reciprocity laws, whose proof is based on MilnorK-theory (see the Appendix). The relation of the 4-function local symbols to the double free loop space of the surface is given by iterated integrals over membranes.

THE PILLAR SWITCHINGS OF MAPPING CLASS GROUPS OF SURFACES

By gluing two copies of surface S0,g+2 along g + 1 holes, we get surface Sg,1. The pillar switching is a self-homeomorphism of Sg,1 which switches two pillars of surfaces by 180° horizontal rotation. We analyze the actions of the pillar switchings on π1Sg,1 and then give concrete expressions of the pillar switchings in terms of standard Dehn twists. The map ψ : Bg → Γg,1 sending the generators of Bg to the pillar switchings on Sg,1 is defined by extending the embedding Bg ↪ Γ0,(g+1),1. We show that this map is injective by analyzing the actions of pillar switchings on π1Sg,1. We also prove that this map induces a trivial homology homomorphism in the stable range. For the proof we use the categorical delooping method. We construct a suitable monoidal 2-functor from tile category to surface category and show that this functor thus induces a map of double loop spaces.

Delooping totalization of a multiplicative operad

The paper shows that under some conditions the totalization of a cosimplicial space obtained from a multiplicative operad is a double loop space of the space of derived morphisms from the associative operad to the operad itself.

Homological stability properties of spaces of rationalJ–holomorphic curves in ℙ2

In a well known work [Se], Graeme Segal proved that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding continuous mapping space through a range of dimensions increasing with degree. In this paper, we address if a similar result holds when other (not necessarily integrable) almost complex structures are put on projective space. We take almost complex structures that are compatible with the underlying symplectic structure. We obtain the following result: the inclusion of the space of based degree k J-holomorphic maps from P^1 to P^2 into the double loop space of P^2 is a homology surjection for dimensions j<3k-2. The proof involves constructing a gluing map analytically in a way similar to McDuff and Salamon in [MS] and Sikorav in [S] and then comparing it to a combinatorial gluing map studied by Cohen, Cohen, Mann, and Milgram in [CCMM].

Homotopy Batalin–Vilkovisky algebras

This paper provides an explicit cofibrant resolution of the operad encoding Batalin–Vilkovisky algebras. Thus it defines the notion of homotopy Batalin–Vilkovisky algebras with the required homotopy properties. To define this resolution, we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin–Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincaré–Birkhoff–Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad \mathcal{BV} . We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian–Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

Homology of iterated loop spaces of homogeneous spaces associated with exceptional Lie groups

We study the mod 2 homology of the double and triple loop spaces of homogeneous spaces associated with exceptional Lie groups. The main computational tools are the Serre spectral sequence for fibrations Ω n + 1 G → Ω n + 1 ( G / H ) → Ω n H for n = 1 , 2 , and the Eilenberg–Moore spectral sequence associated with related fiber squares, which both converge to the same destination space H ∗ ( Ω n ( G / H ) ; F 2 ) . We also develop the generalized Bockstein lemma to determine the higher Bockstein actions.

Classifying spaces for braided monoidal categories and lax diagrams of bicategories

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well-understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street's) geometric nerves.

Braids, mapping class groups, and categorical delooping

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map $$\phi : \beta_{2g} \to \Gamma _{g,1}$$ from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of (Tillmann in Invent Math 130:257–175, 1997). By extending the homomorphism to a functor of monoidal 2-categories, $$\phi$$ is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology.