393 research outputs found
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 of the Thom space of a
spherical fibration 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
has a local splitting replacing the James splitting
of when is a suspension.Comment: Final version, minor change
A new invariant that's a lower bound of LS-category
Let be a simply connected CW-complex of finite type and any
field. A first known lower bound of LS-category is the Toomer
invariant (\cite{Too}). In 's F\'elix et al.
introduced the concept of {\it depth} in algebraic topology and proved the
depth theorem: .
In this paper, we use the Eilenberg-Moore spectral sequence of to
introduce a new numerical invariant, denoted by \textsc{r}(X, \mathbb{K}),
and show that it has the same properties as those of .
When the evaluation map (\cite{FHT88}) is non-trivial and
, we prove that \textsc{r}(X, \mathbb{K})
interpolates and .
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 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 is a Sullivan minimal algebra such that
and , 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
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