Topological invariance of intersection homology without sheaves

In (1) Goresky and MacPherson defined intersection homology groups for triangulable pseudomanifolds and showed they were PL invariants. Then in [2] they generalized these groups to any pseudomanifold and showed they were topological invariants. These groups have generated a great deal of interest. However, [2] is difficult for many mathematicians (including this author) because it requires a familiarity with a great deal of hefty sheaf-theoretic machinery. This is too bad, because the basic ideas behind intersection homology (elucidated in [1]) are very geometric.In this paper we give a sheafless definition of intersection homology groups for an arbitrary stratified set and we give an elementary sheafless proof that they are topological invariants, i.e. independent of the stratification.In doing so, we find some new perversities whose intersection homology groups are topological invariants. Unfortunately, these new perverse intersection homology classes do not seem to intersect with anything (which is probably why they were ignored by Goresky and MacPherson). But in any case these groups are invariants of singular spaces which might be of some interest.     

Intersection Alexander polynomials

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander polynomials that are defined for non-locally-flat knots. We also investigate the relations between the intersection Alexander polynomials of a knot and the intersection and classical Alexander polynomials of the link knots around the singular strata. To facilitate some of these investigations, we introduce spectral sequences for the computation of the intersection homology of certain stratified bundles.     

On smooth Chas-Sullivan loop product in Quillen's geometric complex cobordism of Hilbert manifolds

In [A.J. Baker, C. Ozel, Complex cobordism of Hilbert manifolds with some applications to flag varieties, Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In [C. Ozel, On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups, in preparation], by using Quinn's Transversality Theorem [F. Quinn, Transversal approximation on Banach manifolds, Proc. Sympos. Pure Math. 15 (1970) 213-222], it has been shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It has been shown that the Thom isomorphism in this theory was satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps has been proved. In [M. Chas, D. Sullivan, String topology, math.GT/9911159, 1999], Chas and Sullivan described an intersection product on the homology of loop space LM. In [R.L. Cohen, J.D.S. Jones, A homotopy theoretic realization of string topology, math.GT/0107187, 2001], R. Cohen and J. Jones described a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. In this paper, we will extend this product on cobordism and bordism theories.     

Intersection homology of stratified fibrations and neighborhoods

We derive spectral sequences for the intersection homology of stratified fibrations and approximate tubular neighborhoods in manifold stratified spaces. These neighborhoods include regular neighborhoods in PL stratified spaces.     

The loop orbifold of the symmetric product

Using the loop orbifold of the symmetric product, we give a formula for the Poincaré polynomial of the free loop space of the Borel construction of the symmetric product. We also show that the Chas-Sullivan orbifold product structure in the homology of the free loop space of the Borel construction of the symmetric product induces a ring structure in the homology of the inertia orbifold of the symmetric product. For a general almost complex orbifold, we define a new ring structure on the cohomology of its inertia orbifold which we call the virtual intersection ring. Finally we show that under Poincaré duality in the case of the symmetric product orbifold, both ring structures are isomorphic.     

On the topological Helly theorem

The main result of this paper is the following theorem, related to the missing link in the proof of the topological version of the classical result of Helly: Let be any family of simply connected compact subsets of R² such that for every i,j∈{0,1,2} the intersections Xi∩Xj are path connected and is nonempty. Then for every two points in the intersection there exists a cell-like compactum connecting these two points, in particular the intersection is a connected set.     

Universal (Co) Homology Theories

It is shown that the homology and cohomology theories on separable C*–algebras given by nonstable E–theory are the universal such theories. By specializing to Abelian C*–algebras, we obtain a family of extraordinary Steenrod homology and cohomology theories on pointed compact metric spaces which are the universal such theories in the same way. For each of the extraordinary Steenrod (co)homology theories considered, we describe an –spectrum which represents the theory.     

Examples of cohomology manifolds which are not homologically locally connected

Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions n?3.     

The duality theorem between the t-singular homology and the ws-singular cohomology of orbifolds

In the present paper, we prove that for an n-dimensional compact orbifold with an s-homological orientation, the duality of the ws-singular cohomology group and the t-singular homology group holds. The key tools are “the t-modification of the cap product” for giving the duality homomorphism and “the Convex Suborbifold Theorem” for extending the local duality isomorphism to the global one. The duality theorem proved in the present paper is a naturally required consequence of the preceding works of the authors.     

On the universal coefficients formula for shape homology

In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.     

REMARKS ON UNIVERSAL COEFFICIENT THEOREMS FOR GENERALIZED HOMOLOGY THEORIES

Abstract

New proofs of universal coefficient theorems for generalized homology theories (cf. ∮ 2, ∮ 3) including L. G. Brown's result, relating Brown-Douglas-Fillmore's Ext (X) with complex K-theory are presented. They are all based on a theorem asserting the existence of a chain functor for a generalized homology theory (cf. ∮ 1), which was originally designed for the construction of strong homology theories on strong shape categories.     

On the vanishing of (co)homology over local rings

Considering modules of finite complete intersection dimension over commutative Noetherian local rings, we prove (co)homology vanishing results in which we assume the vanishing of nonconsecutive (co)homology groups. In fact, the (co)homology groups assumed to vanish may be arbitrarily far apart from each other.     

Milnor–Thurston homology groups of the Warsaw Circle

Milnor–Thurston homology theory is a construction of homology theory that is based on measures. It is known to be equivalent to singular homology theory in case of manifolds and complexes. Its behaviour for non-tame spaces is still unknown. This paper provides results in this direction. We prove that Milnor–Thurston homology groups for the Warsaw Circle are trivial except for the zeroth homology group which is uncountable-dimensional. Additionally, we prove that the zeroth homology group is non-Hausdorff for this space with respect a natural topology that was proposed by Berlanga.     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.     

On the monodromy action on Milnor fibers of graphic arrangements

We analyze the monodromy action, over the rationals, on the first homology group of the Milnor fiber, for arbitrary subarrangements of Coxeter arrangements. We propose a combinatorial formula for the monodromy action, involving Aomoto complexes in positive characteristic. We verify the formula, in cases A, B and D.     

Strong homology of inverse systems. III

In previous parts I and II of this paper [4] strong homology groups of inverse systems were introduced and studied. In this part III of the paper we define strong homology groups of inverse systems of pairs and show that they have suitable exactness and excision properties. As a consequence of these results the Steenrod-Sitnikov homology [1] for pairs (X,A), where X is a paracompact space and A is a closed subset of X, is exact and satisfies the excision axiom.     

The t-singular homology of orbifolds

For an orbifold M we define a new homology group, called t-singular homology group t-Hq(M) by using singular simplicies intersecting ‘transversely’ with ΣM. The rightness of this homology group is ensured by the facts that the 1-dimensional homology group t-H₁(M) is isomorphic to the abelianization of the orbifold fundamental group π₁(M,x₀). If M is a manifold, t-Hq(M) coincides with the usual singular homology group. We prove that it is a ‘b-homotopy’ invariant of orbifolds and develop many algebraic tools for the calculations. Consequently we calculate the t-singular homology groups of several orbifolds.     

Twisted Novikov homology of complex hypersurface complements

We study the twisted Novikov homology of the complement of a complex hypersurface in general position at infinity. We give a self‐contained topological proof of the vanishing (except possibly in the middle degree) of the twisted Novikov homology groups associated to positive cohomology classes of degree one defined on the complement.     

Lifting problems and transgression for non-abelian gerbes

We discuss lifting and reduction problems for bundles and gerbes in the context of a Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen’s long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in non-abelian cohomology. This transgression map relates the degree one non-abelian cohomology of a smooth manifold (represented by non-abelian gerbes) with the degree zero non-abelian cohomology of the free loop space (represented by principal bundles). We prove several properties for this transgression map. For instance, it reduces–in case of a Lie 2-group with a single object–to the ordinary transgression in ordinary cohomology. We describe applications of our results to string manifolds: first, we obtain a new comparison theorem for different notions of string structures. Second, our transgression map establishes a direct relation between string structures and spin structures on the loop space.