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Several stochastic processes related to transient Lévy processes with potential densities Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional properties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.

We study the $E_2$ -algebra $\\Lambda \\mathfrak {M}_{*,1}:= \\coprod _{g\\geqslant 0}\\Lambda \\mathfrak {M}_{g,1}$ consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion $\\Omega B\\Lambda \\mathfrak {M}_{*,1}$ : it is the product of $\\Omega ^{\\infty }\\mathbf {MTSO}(2)$ with a certain free $\\Omega ^{\\infty }$ -space depending on the family of all boundary-irreducible mapping classes in all mapping class groups $\\Gamma _{g,n}$ with $g\\geqslant 0$ and $n\\geqslant 1$ .

Anick spaces are closely connected with both EHP sequences and the study of torsion exponents. In addition they refine the secondary suspension and enter unstable periodicity. In this work we describe their

For a partially multiplicative quandle (PMQ) ${\\mathcal {Q}}$ we consider the topological monoid $\\mathring {\\mathrm {HM}}({\\mathcal {Q}})$ of Hurwitz spaces of configurations in the plane with local monodromies in ${\\mathcal {Q}}$. We compute the group completion of $\\mathring {\\mathrm {HM}}({\\mathcal {Q}})$: it is the product of the (discrete) enveloping group ${\\mathcal {G}}({\\mathcal {Q}})$ with a component of the double loop space of the relative Hurwitz space $\\mathrm {Hur}_+([0,1]^2,\\partial [0,1]^2;{\\mathcal {Q}},G)_{\\mathbb {1}}$; here $G$ is any group giving rise, together with ${\\mathcal {Q}}$, to a PMQ–group pair. Under the additional assumption that ${\\mathcal {Q}}$ is finite and rationally Poincaré and that $G$ is finite, we compute the rational cohomology ring of $\\mathrm {Hur}_+([0,1]^2,\\partial [0,1]^2;{\\mathcal {Q}},G)_{\\mathbb {1}}$.

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of two-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle that covers the fusion of loops, i.e. the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish novel relations between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.

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.

We generalise the fold map for the wedge sum and use this to give a loop space decomposition of topological spaces with a high degree of symmetry. This is applied to polyhedral products to give a loop space decomposition of polyhedral products associated to families of graphs.

The cohomology algebra of the space H ∗ ( X ) defines neither cohomology modules of the loop space H ∗ (Ω X ) nor cohomologies of the free loop space H ∗ (Λ X ). But by the author’s minimality theorem, there exists a structure of A ( ∞ )-algebra ( H ∗ ( X ) , { m i }) on H ∗ ( X ), which determines H ∗ (Ω X ). Here will be shown that the same A ( ∞ )-algebra ( H ∗ ( X ) , { m i }) determines also cohomology modules H ∗ (Λ X ).