Abstract:
This note provides a reference for some properties of the Thom spectrum $M\xi$ over $\Omega\Sigma\CPi$. Some of this material is used in recent work of Kitchloo and Morava. We determine the $M\xi$-cohomology of $\CPi$ and show that $M\xi^*(\CPi)$ injects into power series over the algebra of non-symmetric functions. We show that $M\xi$ gives rise to a commutative formal group law over the non-commutative ring $\pi_*M\xi$. We also discuss how $M\xi$ and some real and quaternionic analogues behave with respect to spectra that are related to these Thom spectra by splittings and by maps.

Abstract:
Let X be a simply connected space and k a commutative ring. Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild cohomology of the singular chains on the pointed loop space HH^{*}S_*(\Omega X) is isomorphic to the free loop space cohomology H^{*}(X^{S^{1}}). We proved that this isomorphism is compatible with both the cup product on HH^{*}S_*(\Omega X) and on H^{*}(X^{S^{1}}). In particular, we explicit the algebra H^{*}(X^{S^{1}}) when X is a suspended space, a complex projective space or a finite CW-complex of dimension p such that \frac {1}{(p-1)!}\in k.

Abstract:
Let $X$ be a simply connected space and $\Bbb K$ be any field. The normalized singular cochains $N^*(X; {\Bbb K})$ admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology $HH_* N^*X$ of the space $X$. We prove that, endowed with this product, $HH_*N^*X$ is isomorphic to the cohomology algebra of the free loop space of $X$ with coefficients in $\Bbb K$. We also show how to construct a simpler Hochschild complex which allows direct computation.

Abstract:
We study natural subalgebras Ch_E(G) of group cohomology defined in terms of infinite loop spaces E and give representation theoretic descriptions of those based on QS^0 and the Johnson-Wilson theories E(n). We describe the subalgebras arising from the Brown-Peterson spectra BP and as a result give a simple reproof of Yagita's theorem that the image of BP^*(BG) in H^*(BG;F_p) is F-isomorphic to the whole cohomology ring; the same result is shown to hold with BP replaced by any complex oriented theory E with a map of ring spectra from E to HF_p which is non-trivial in homotopy. We also extend the constructions to define subalgebras of H^*(X;F_p) for any space X; when X is finite we show that the subalgebras Ch_{E(n)}(X) give a natural unstable chromatic filtration of H^*(X;F_p).

Abstract:
Let X be a 1-connected compact space such that the algebra H*(X;Z/2) is generated by one single element. We compute the cohomology of the free loop space H*(LX;Z/2) including the Steenrod algebra action. When X is a projective space CP^n, HP^n, the Cayley projective plane CaP^2 or a sphere S^m we obtain a splitting result for integral and mod two cohomology of the suspension spectrum of LX_+. The splitting is in terms of the suspension spectrum of X_+ and the Thom spaces of the q-fold Whitney sums of the tangent bundle over X for non negative integers q.

Abstract:
The goal of this article is to make explicit a structured complex whose homology computes the cohomology of the p-profinite completion of the n-fold loop space of a sphere of dimension d=n-m

Abstract:
We study the mod-2 cohomology spectral sequence arising from delooping the Bousfield-Kan cosimplicial space giving the 2-nilpotent completion of a connective spectrum $X$. Under good conditions its $E_{2}$-term is computable as certain non-abelian derived functors evaluated at $H^*(X)$ as a module over the Steenrod algebra, and it converges to the cohomology of $\Omega^\infty X$. We provide general methods for computing the $E_{2}$-term, including the construction of a multiplicative spectral sequence of Serre type for cofibration sequences of simplicial commutative algebras. Some simple examples are also considered; in particular, we show that the spectral sequence collapses at $E_{2}$ when $X$ is a suspension spectrum.

Abstract:
We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Chern character form, a loop space refinement of the twisted Chern character form, which represent classes in the completed periodic exotic twisted $S^1$-equivariant cohomology of $LZ$. We establish a localisation theorem for the completed periodic exotic twisted $S^1$-equivariant cohomology for loop spaces and apply it to establish T-duality in a background flux in type II String Theory from a loop space perspective.

Abstract:
In this paper we calculate the integral Pontrjagin homology ring of the based loop space on some generalised symmetric spaces with a toral stationary subgroup. In the Appendix we show that the method can be applied to other type generalised symmetric spaces as well.

Abstract:
Let X be a compact connected Riemann surface of genus at least two. We compute the Chen--Ruan cohomology ring of the moduli space of stable PSL(2, C)--bundles of nontrivial second Stiefel--Whitney class over X.