57 research outputs found
Natural operations in Intersection Cohomology
Eilenberg-MacLane spaces, that classify the singular cohomology groups of
topological spaces, admit natural constructions in the framework of simplicial
sets. The existence of similar spaces for the intersection cohomology groups of
a stratified space is a long-standing open problem asked by M. Goresky and R.
MacPherson. One feature of this work is a construction of such simplicial sets.
From works of R. MacPherson, J. Lurie and others, it is now commonly accepted
that the simplicial set of singular simplices associated to a topological space
has to be replaced by the simplicial set of singular simplices that respect the
stratification. This is encoded in the category of simplicial sets over the
nerve of the poset of strata. For each perversity, we define a functor from it,
with values in the category of cochain complexes over a commutative ring. This
construction is based upon a simplicial blow up and the associated cohomology
is the intersection cohomology as it was defined by M. Goresky and R.
MacPherson in terms of hypercohomology of Delignes's sheaves. This functor
admits an adjoint and we use it to get classifying spaces for intersection
cohomology. Natural intersection cohomology operations are understood in terms
of intersection cohomology of these classifying spaces. As in the classical
case, they form infinite loop spaces. In the last section, we examine the depth
one case of stratified spaces with only one singular stratum. We observe that
the classifying spaces are Joyal's projective cones over classical
Eilenberg-MacLane spaces. We establish some of their properties and conjecture
that, for Goresky and MacPherson perversities, all intersection cohomology
operations are induced by classical ones
A bordism approach to string topology
Using intersection theory in the context of Hilbert manifolds and geometric
homology we show how to recover the main operations of string topology built by
M. Chas and D. Sullivan. We also study and build an action of the homology of
reduced Sullivan's chord diagrams on the singular homology of free loop spaces,
extending previous results of R. Cohen and V. Godin and unifying part of the
rich algebraic structure of string topology as an algebra over the prop of
these reduced diagrams. Some of these operations are extended to spaces of maps
from a sphere to a compact manifold
Mixed Hodge structures on the intersection homotopy type of complex varieties with isolated singularities
A homotopical treatment of intersection cohomology recently developed by
Chataur-Saralegui-Tanr\'e associates a "perverse algebraic model" to every
topological pseudomanifold, extending Sullivan's presentation of rational
homotopy to intersection cohomology. In this context, there is a notion of
"intersection-formality", measuring the vanishing of Massey products in
intersection cohomology. In the present paper, we study the perverse algebraic
model of complex projective varieties with isolated singularities. We endow
such invariant with natural mixed Hodge structures. This allows us to prove
some intersection-formality results for large families of complex projective
varieties, such as isolated surface singularities and varieties of arbitrary
dimension with ordinary isolated singularities
Fibrewise nullification and the cube theorem
Our aim is to construct fibrewise localizations in model categories. For
pointed spaces, the general idea is to decompose the total space of a fibration
as a diagram over the category of simplices of the base and replace it by the
localized diagram. This of course is not possible in an arbitrary category. We
have thus to adapt another construction which heavily depends on Mather's cube
theorem. Working with model categories in which the cube theorem holds, we
characterize completely those who admit a fibrewise nullification. As an
application we get fibrewise plus-construction and fibrewise Postnikov sections
for algebras over an operad
Rational homotopy of complex projective varieties with normal isolated singularities
Let X be a complex projective variety of dimension n with only isolated
normal singularities. In this paper we prove, using mixed Hodge theory, that if
the link of each singular point of X is (n-2)-connected, then X is a formal
topological space. This result applies to a large class of examples, such as
normal surface singularities, varieties with ordinary multiple points,
hypersurfaces with isolated singularities and more generally, complete
intersections with isolated singularities. We obtain analogous results for
contractions of subvarieties.Comment: Final version to appear in Forum. Math. Exposition improved thanks to
the referees comment
Homology of spaces of regular loops in the sphere
In this paper we compute the singular homology of the space of immersions of
the circle into the -sphere. Equipped with Chas-Sullivan's loop product
these homology groups are graded commutative algebras, we also compute these
algebras. We enrich Morse spectral sequences for fibrations of free loop spaces
together with loop products, this offers some new computational tools for
string topology.Comment: 32 pages, 12 figure
Steenrod squares on Intersection cohomology and a conjecture of M. Goresky and W. Pardon
We prove a conjecture raised by M. Goresky and W. Pardon, concerning the
range of validity of the perverse degree of Steenrod squares in intersection
cohomology. This answer turns out of importance for the definition of
characteristic classes in the framework of intersection cohomology.
For this purpose, we present a construction of -products on
the cochain complex, built on the blow-up of some singular simplices and
introduced in a previous work. We extend to this setting the classical
properties of the associated Steenrod squares, including Adem and Cartan
relations, for any generalized perversities. In the case of a pseudomanifold,
we prove that our definition coincides with M. Goresky's definition.
Several examples of concrete computation of perverse Steenrod squares are
given, including the case of isolated singularities and, more especially, we
describe the Steenrod squares on the Thom space of a vector bundle, in function
of the Steenrod squares of the basis and the Stiefel-Whitney classes. We detail
also an example of a non trivial square, $\sq^2\colon H_{\ bar{p}}\to H_{\
bar{p}+2}H_{2\
bar{p}}$, showing the interest of the Goresky and Pardon's conjecture.Comment: Correction of some misprint
Operadic Hochschild chain complex and free loop spaces
We construct for any algebra over an operad an Hochschild chain complex. In
the case of the singular cochain complex of a topological space, considered as
a commutative algebra up to homotopy, we show that this complex computes the
singular cohomology of the free loop space over this topological space.Comment: 13 page
Frobenius Rational Loop Algebra
Recently R. Cohen and V. Godin have proved that the homology of the free loop
space of a closed oriented manifold with coefficients in a field has the
structure of a Frobenius algebra without counit. In this short note we prove
that when the characteristic of the field is zero and when the manifold is
1-connected the algebraic structure depends only on the rational homotopy type
of the manifold. We build an algebraic model and use it to do some
computations.Comment: 14 page
Poincar\'e duality with cap products in intersection homology
For having a Poincar\'e duality via a cap product between the intersection
homology of a paracompact oriented pseudomanifold and the cohomology given by
the dual complex, G. Friedman and J. E. McClure need a coefficient field or an
additional hypothesis on the torsion. In this work, by using the classical
geometric process of blowing-up, adapted to a simplicial setting, we build a
cochain complex which gives a Poincar\'e duality via a cap product with
intersection homology, for any commutative ring of coefficients. We prove also
the topological invariance of the blown-up intersection cohomology with compact
supports in the case of a paracompact pseudomanifold with no codimension one
strata.
This work is written with general perversities, defined on each stratum and
not only in function of the codimension of strata. It contains also a tame
intersection homology, suitable for large perversities.Comment: A new subsection has been adde
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