This article is devoted to the study of smooth desingularization, which are customary employed in the definition of De Rham Intersection Cohomology with differential forms [12]. In this paper we work with the category of Thom-Mather simple spaces [10], [14]. We construct a functor which sends each Thom-Mather simple space into a smooth manifold called its primary unfolding. Hence we prove that ...

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes’ basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic Thom conjecture: a symplectic surface in a symplectic fourmanifold is genus-...

Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral cohomology when restricted to the handlebody subgroup. This is a special case of a more general theorem valid in all dimensions: universal characteristic classes ma...

At the 1958 International Congress of Mathematicians in Edinburgh, René Thom received one of two Field Medals for his development of cobordism. In his citation [11] Heinz Hopf described the definition of cobordism as one of those elementary and apparently trivial constructions which can hardly be expected to yield significant results. He compares this with Hurewicz’s definition of homotopy grou...

To each subset I of {1, . . . , k} associate an integer r(I). Denote by X the collection of those n × k matrices for which the rank of a union of columns corresponding to a subset I is r(I), for all I. We study the equivariant cohomology class represented by the Zariski closure Y = X. This class is an invariant of the underlying matroid structure. Its calculation incorporates challenges similar...

We compute the mod 2 homology of spin mapping class groups in the stable range. In earlier work [G] we computed the stable mod p homology of the oriented mapping class group, and the methods and results here are very similar. The forgetful map from the spin mapping class group to the oriented mapping class groups induces a homology isomorphism for odd p but for p = 2 it is far from being an iso...

We give a Pontryagin-Thom type construction for −1-codimensional fold maps. We obtain results about cobordism and bordism groups of fold maps of oriented (n+1)-dimensional manifolds into n-dimensional manifolds in an analogous way to the Pontryagin-Thom type construction for positive codimensional singular maps.

In the late 1940s N. Steenrod posed the following problem, which is now familiar as the problem on realisation of cycles. For a given homology class z ∈ Hn(X;Z), do there exist an oriented manifold N and a mapping f : N → X such that f∗[N ] = z? A famous theorem of R. Thom claims that each integral homology class is realisable in sense of Steenrod with some multiplicity. A classical problem is ...

Dold-Thom functors are generalizations of infinite symmetric products, where integer multiplicities of points are replaced by composable elements of a partial abelian monoid. It is well-known that for any connective homology theory, the machinery of Γ-spaces produces the corresponding linear Dold-Thom functor. In this note we show how to obtain such functors directly

We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing of vector bundle-valued fermions yields representative the Thom class. This constructs complexified string orientation cohomology, determines pushfoward families rational manifolds. A second pu...