Related papers: Stable Pontryagin-Thom construction for proper map…
In a preceding article the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above…
Thom polynomials are universal cohomological obstructions to the appearance of singularities of given types in differentiable maps. As an application, various invariants of immersions have been expressed in terms of singularities of their…
We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples,…
The well-known theory of Pontryagin duality provides a strong connection between the homology and cohomology theories of a profinite group in appropriate categories. A construction for taking the `profinite direct sum' of an infinite family…
Let $X$ and $Y$ be irreducible normal projective varieties, of same dimension, defined over an algebraically closed field, and let $f : Y \rightarrow X$ be a finite generically smooth morphism such that the corresponding homomorphism…
Together with F. Morel, we have constructed in \cite{CR, Cobord1, Cobord2} a theory of {\em algebraic cobordism}, an algebro-geometric version of the topological theory of complex cobordism. In this paper, we give a survey of the…
An equivariant Thom isomorphism theorem in operator K-theory is formulated and proven for infinite rank Euclidean vector bundles over finite dimensional Riemannian manifolds. The main ingredient in the argument is the construction of a…
We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of…
We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausen's…
This paper identifies the homotopy theories of topological stacks and orbispaces with unstable global homotopy theory. At the same time, we provide a new perspective by interpreting it as the homotopy theory of `spaces with an action of the…
Let ${\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, smooth curves. For a family $C\to S$ of such curves over a connected base and a geometric point $\xi$ on $S$, the associated monodromy representation is…
(This is an updated version; following an idea of Voevodsky, we have strengthened our results so all of them apply to one form of motivic homotopy theory). We give two general constructions for the passage from unstable to stable homotopy…
Ren\'e Thom's remarkable and far-reaching concept of transversality has found numerous powerful applications. Most importantly, it allowed Thom to develop cobordism theory, which led to a piercing insight into the topology of smooth…
Let $\mathcal{T}$ be a $k$-linear triangulated category. The space of Bridgeland stability conditions on $\mathcal{T}$, denoted by $\mathrm{Stab}(\mathcal{T})$, forms a complex manifold. In this paper, we introduce an equivalence relation…
Let $f,g:X \to Y$ be continuous mappings. We say that $f$ is topologically equivalent to $g$ if there exist homeomorphisms $\Phi : X\to X$ and $\Psi: Y\to Y$ such that $\Psi\circ f\circ \Phi=g.$ Let $X,Y$ be complex smooth irreducible…
In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is…
We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in…
A map Y -> P^n is determined by a line bundle quotient of (O_Y)^{n+1}. In this paper, we generalize this description to the case of maps from Y to an arbitrary smooth toric variety. The data needed to determine such a map consists of a…
We introduce a new map between configuration spaces of points in a background manifold - the replication map - and prove that it is a homology isomorphism in a range with certain coefficients. This is particularly of interest when the…
We compute the cohomology ring of a generalised type of configuration space of points in $\mathbb{R}^r$. This configuration space is indexed by a graph. In the case the graph is complete the result is known and it is due to Arnold and…