Dold Thom Theorem Research Articles

Homotopy truncations of homotopically stratified spaces

Intersection homology of Goresky and MacPherson can be defined from the Deligne sheaf, obtained from truncations of complexes of sheaves. As intersection homology is not the homology of a particular space, the search for a family of spaces whose homologies have properties analogous to intersection homology has developed. For some stratified spaces, M. Banagl has introduced such a family by using a topological truncation: the original link is replaced by a truncation of its homological Moore resolution. In this work, we study the dual approach in the Eckmann-Hilton sense: we consider the stratified space obtained by replacing the original link by a Postnikov approximation. The main result is that our construction restores the space constructed by Gajer to establish an intersection Dold-Thom theorem. We are conducting this study within the general framework of Quinn’s homotopically stratified spaces.

Infinite symmetric products of rational algebras and spaces

We show that the infinite symmetric product of a connected graded-commutative algebra over ℚ is naturally isomorphic to the free graded-commutative algebra on the positive degree subspace of the original algebra. In particular, the infinite symmetric product of a connected commutative (in the usual sense) graded algebra over ℚ is a polynomial algebra. Applied to topology, we obtain a quick proof of the Dold–Thom theorem in rational homotopy theory for connected spaces of finite type. We also show that finite symmetric products of certain simple free graded-commutative algebras are free; this allows us to determine minimal Sullivan models for finite symmetric products of complex projective spaces.

The Dold–Thom theorem via factorization homology

We give a new proof of the classical Dold–Thom theorem using factorization homology. Our method is direct and conceptual, avoiding the Eilenberg–Steenrod axioms entirely in favor of a more general geometric argument.

Nash equilibria via duality and homological selection

Given a multifunction from X to the k-fold symmetric product Sym k (X), we use the Dold–Thom theorem to establish a homological selection theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash equilibria are traditionally multilinear in the mixed strategies. The main aim of this paper is to relax the hypothesis of multilinearity. We use basic intersection theory, Poincaré duality in addition to the Dold–Thom theorem.

Symmetric powers do not stabilize

We discuss the stabilization of symmetric products Sym^n(X) of a smooth projective variety X in the Grothendieck ring of varieties. For smooth projective surfaces X with non-zero h^0(X, \omega_X), these products do not stabilize; we conditionally show that they do not stabilize in another related sense, in response to a question of R. Vakil and M. Wood. There are analogies between such stabilization, the Dold-Thom theorem, and the analytic class number formula. Finally, we discuss Hodge-theoretic obstructions to the stabilization of symmetric products, and provide evidence for these obstructions in terms of a relationship between the Newton polygon of a certain motivic zeta function associated to a curve, and its Hodge polygon.

Free abelian topological groups and collapsing maps

We show that for every pair (X,Y) of ANR compacta, Y⊂X, the free abelian topological group applied to a collapsing map q:X→X/Y produces a locally trivial bundle A(q):A(X)→A(X/Y) with the fiber A(Y). As a result we obtain a short proof of the classical Dold–Thom theorem which states that H˜n(X)=πn(A(X)) for all complexes X.

Stable equivariant abelianization, its properties, and applications

Let G be a finite group. For a based G-space X and a Mackey functor M, a topological Mackey functor X ⊗ ˜ M is constructed, which will be called the stable equivariant abelianization of X with coefficients in M. When X is a based G-CW complex, X ⊗ ˜ M is shown to be an infinite loop space in the sense of G -spaces. This gives a version of the RO ( G ) -graded equivariant Dold–Thom theorem. Applying a variant of Elmendorf's construction, we get a model for the Eilenberg–Mac Lane spectrum HM. The proof uses a structure theorem for Mackey functors and our previous results.

Topological Abelian Groups and Equivariant Homology

We prove an equivariant version of the Dold–Thom theorem by giving an explicit isomorphism between Bredon–Illman homology and equivariant homotopical homology π*(F G (X, L)), where G is a finite group and L is a G-module. We use the homotopical definition to obtain several properties of this theory and we do some calculations.

A functor converting equivariant homology to homotopy

In this paper, an equivariant version of the classical Dold–Thom theorem is proved. Let G be a finite group, X a G-space, and k a covariant coefficient system on G. Then a topological abelian group 𝒢X ⊗Gℱ k is constructed by the coend construction. For a G-CW complex X, it is proved that there is a natural isomorphism π i ( 𝒢 X ⊗ G ℱ k ) ≅ H i G ( X k ), where the right-hand side is the Bredon equivariant homology of X with coefficients in k. At the end, several examples of this result are presented.

A note on the equivariant Dold–Thom theorem

In this note we prove a version of the classical Dold–Thom theorem for the RO( G)-graded equivariant homology functors H ∗ G(−; M ̄ ) , where G is a finite group, M is a discrete Z[G] -module, and M̱ is the Mackey functor associated to M. In the case where M= Z with the trivial G-action, our result says that, for a G-CW-complex X, and for a finite dimensional G-representation V, there is a natural isomorphism [S V, Z 0(X)] G≅H V G(X; Z ), where Z 0(X) denotes the free abelian group on X.

The intersection Dold-Thom theorem

The intersection Dold-Thom theorem

Formes différentielles non commutatives et cohomologie à coefficients arbitraires

The purpose of the paper is to promote a new definition of cohomology, using the theory of non commutative differential forms, introduced already by Alain Connes and the author in order to study the relation between K K -theory and cyclic homology. The advantages of this theory in classical Algebraic Topology are the following: A much simpler multiplicative structure, where the symmetric group plays an important role. This is important for cohomology operations and the investigation of a model for integral homotopy types (Formes différentielles non commutatives et opérations de Steenrod, Topology, to appear). These considerations are of course related to the theory of operads. A better relation between de Rham cohomology (defined through usual differential forms on a manifold) and integral cohomology, thanks to a "non commutative integration". A new definition of Deligne cohomology which can be generalized to manifolds provided with a suitable filtration of their de Rham complex. In this paper, the theory is presented in the framework of simplicial sets. With minor modifications, the same results can be obtained in the topological category, thanks essentially to the Dold-Thom theorem (Formes topologiques non commutatives, Ann. Sci. Ecole Norm. Sup., to appear). A summary of this paper has been presented to the French Academy: CR Acad. Sci. Paris 316 (1993), 833-836.

Filtrations on the homology of algebraic varieties

Introduction Questions and speculations Abelian monoid varieties Chow varieties and Lawson homology Correspondences and Lawson homology Multiplication of algebraic cycles Operations in Lawson homology Filtrations Appendix A. Mixed Hodge structures, homology, and cycle classes Appendix B. Trace maps and the Dold-Thom theorem Appendix Q. On the group completion of a simplicial monoid Bibliography.

Completions and fibrations for topological monoids

We show that, for a certain class of topological monoids, there is a homotopy equivalence between the homotopy theoretic group completion M + {M^ + } of a monoid M M in that class and the topologized Grothendieck group M ~ \tilde M associated to M M . The class under study is broad enough to include the Chow monoids effective cycles associated to a projective algebraic variety and also the infinite symmetric products of finite CW {\text {CW}} -complexes. We associate principal fibrations to the completions of pairs of monoids, showing the existence of long exact sequences for the naïve approach to Lawson homology [Fri91, LF91a]. Another proof of the Eilenberg-Steenrod axioms for the functors X ↦ S P ~ ( X ) X \mapsto {\tilde {SP}}(X) in the category of finite CW {\text {CW}} -complexes (Dold-Thom theorem [DT56]) is obtained.

Hecke Functors and the Equivariant Dold-Thom Theorem

We recall that the Dold-Thom theorem [3] asserts that the weak homotopy type cf a topological abelian group is determined by its homotopy groups, and hence the homotopy category cf topological abelian groups with respect to weak equivalences is equivalent to the category cf graded abelian groups. In this note we consider the equivariant version. We restrict ourself to the case cf finite group actions. Let G be a finite group, and let k be a commutative ring with unit. A topological &[G]module is a topological abelian group M as well as a &[G]-module such that the bilinear map

The relative Dold-Thom theorem

The relative Dold-Thom theorem