On the cohomology of loop spaces for some Thom spaces

In this paper we identify conditions under which the cohomology H^*(\Omega M\xi;\k) for the loop space $\Omega M\xi$ of the Thom space $M\xi$ of a spherical fibration $\xi\downarrow B$ can be a polynomial ring. We use the Eilenberg-Moore spectral sequence which has a particularly simple form when the Euler class e(\xi)\in H^n(B;\k) vanishes, or equivalently when an orientation class for the Thom space has trivial square. As a consequence of our homological calculations we are able to show that the suspension spectrum $\Sigma^\infty\Omega M\xi$ has a local splitting replacing the James splitting of $\Sigma\Omega M\xi$ when $M\xi$ is a suspension.Comment: Final version, minor change

A new invariant that's a lower bound of LS-category

Let $X$ be a simply connected CW-complex of finite type and $\mathbb{K}$ any field. A first known lower bound of LS-category $cat(X)$ is the Toomer invariant $e_{\mathbb{K}} (X)$ (\cite{Too}). In $1980$'s F\'elix et al. introduced the concept of {\it depth} in algebraic topology and proved the depth theorem: $depth (H_*(\Omega X, \mathbb{K})) \leq cat(X)$. In this paper, we use the Eilenberg-Moore spectral sequence of $X$ to introduce a new numerical invariant, denoted by \textsc{r}(X, \mathbb{K}), and show that it has the same properties as those of $e_{\mathbb{K}} (X)$. When the evaluation map (\cite{FHT88}) is non-trivial and $char(\mathbb{K})\not = 2$, we prove that \textsc{r}(X, \mathbb{K}) interpolates $depth(H_*(\Omega X, \mathbb{K}))$ and $e_{\mathbb{K}} (X)$. Hence, we obtain an improvement of L. Bisiaux theorem (\cite{Bis99}) and then of the depth theorem. Motivated by these results, we associate to any commutative differential graded algebra $(A,d)$, a purely algebraic invariant \textsc{r}(A,d) and, via the theory of minimal models, we relate it with our previous topological results. In particular, if $(\Lambda V,d)$ is a Sullivan minimal algebra such that $d=\sum_{i\geq k}d_i$ and $d_i(V)\subseteq \Lambda ^iV$, a greater lower bound is obtained, namely e_0(\Lambda V, d)\geq \textsc{r}(\Lambda V, d) + (k-2).Comment: 21 page

A spectral sequence for string cohomology

Let X be a 1-connected space with free loop space LX. We introduce two spectral sequences converging towards H^*(LX;Z/p) and H^*((LX)_hT;Z/p). The E2-terms are certain non Abelian derived functors applied to H^*(X;Z/p). When H^*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.Comment: 38 page

Duality in algebra and topology

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to the more general rings that come up in homotopy theory. Amongst the rings we work with are the differential graded ring of cochains on a space, the differential graded ring of chains on the loop space, and various ring spectra, e.g., the Spanier-Whitehead duals of finite spectra or chromatic localizations of the sphere spectrum. Maybe the most important contribution of this paper is the conceptual framework, which allows us to view all of the following dualities: Poincare duality for manifolds, Gorenstein duality for commutative rings, Benson-Carlson duality for cohomology rings of finite groups, Poincare duality for groups, Gross-Hopkins duality in chromatic stable homotopy theory, as examples of a single phenomenon. Beyond setting up this framework, though, we prove some new results, both in algebra and topology, and give new proofs of a number of old results

Complete intersections and mod p cochains

We give homotopy invariant definitions corresponding to three well known properties of complete intersections, for the ring, the module theory and the endomorphisms of the residue field, and we investigate them for the mod p cochains on a space, showing that suitable versions of the second and third are equivalent and that the first is stronger. We are particularly interested in classifying spaces of groups, and we give a number of examples. This paper follows on from arXiv:0906.4025 which considered the classical case of a commutative ring and arXiv:0906.3247 which considered the case of rational homotopy theory.Comment: To appear in AG