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CW Type of Inverse Limits and Function Spaces
, 708
"... Hspace exponent, EilenbergMacLane space. ..."
Stable homotopical algebra and Gammaspaces
, 1999
"... this paper we advertise the category of #spaces as a convenient framework for doing `algebra' over `rings' in stable homotopy theory. #spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (1)conne ..."
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this paper we advertise the category of #spaces as a convenient framework for doing `algebra' over `rings' in stable homotopy theory. #spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (1)connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for #spaces. The study of `rings, modules and algebras' based on #spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their EilenbergMacLane spectra counterparts. There are other settings for ring spectra, most notably the Smodules and Salgebras of [EKMM] and the symmetric spectra of [HSS], each of these with its own advantages and disadvantages. We believe that one advantage of the # space approach is its simplicity. The definitions of the stable equivalences, the smash product and the `rings' (which we call Gammarings) are given on a few pages. Another feature is that #spaces nicely reflect the idea that spectra are a homotopical generalization of abelian groups, that the smash product generalizes the tensor product and that algebras over the sphere spectrum generalize classical rings. There is an EilenbergMacLane functor H which embeds the category of simplicial abelian groups as a full subcategory of the category of #spaces. The embedding has a left adjoint, left inverse which on cofibrant objects models spectrum homology. Similarly, simplicial rings embed fully faithfully into Gammarings. We give a quick proof (see Section 4) ...
A History of Duality in Algebraic Topology
"... This paper became the starting point of investigations of homology for more general spaces than merely finite complexes or open subsets of R ..."
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This paper became the starting point of investigations of homology for more general spaces than merely finite complexes or open subsets of R
FUNCTOR CONVERTING EQUIVARIANT HOMOLOGY TO HOMOTOPY
, 2006
"... Abstract. In this paper, we prove an equivariant version of the classical DoldThom theorem. Let G be a finite group, X a Gspace, and k a covariant coefficient system on G. We construct a topological abelian group GX ⊗B G k by the coend construction. We then prove that for a GCW complex X, πi(GX ⊗ ..."
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Abstract. In this paper, we prove an equivariant version of the classical DoldThom theorem. Let G be a finite group, X a Gspace, and k a covariant coefficient system on G. We construct a topological abelian group GX ⊗B G k by the coend construction. We then prove that for a GCW complex X, πi(GX ⊗B G k) ∼ = HG i (X; k), where the right hand side is the Bredon equivariant homology of X with coefficients in k. At the end we present several examples of this result. Contents
Lagrangian Dynamics on an infinitedimensional torus; a Weak KAM theorem
"... The space L 2 (0, 1) has a natural Riemannian structure on the basis of which we introduce an L 2 (0, 1)–infinite dimensional torus T. For a class of Hamiltonians defined on its cotangent bundle we establish existence of a viscosity solution for the cell problem on T or, equivalently, we prove a Wea ..."
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The space L 2 (0, 1) has a natural Riemannian structure on the basis of which we introduce an L 2 (0, 1)–infinite dimensional torus T. For a class of Hamiltonians defined on its cotangent bundle we establish existence of a viscosity solution for the cell problem on T or, equivalently, we prove a Weak KAM theorem. As an application, we obtain existence of absolute actionminimizing solutions of prescribed rotation number for the onedimensional nonlinear Vlasov system with periodic potential. 1
TEN TOPOLOGIES FOR 1x7
, 1963
"... THE study of topologies on X x Y is motivated by some outstanding deficiencies of the cartesian, that is the usual, topology on the product of spaces. (Throughout this paper all spaces will be assumed to be Hausdorff.) ..."
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THE study of topologies on X x Y is motivated by some outstanding deficiencies of the cartesian, that is the usual, topology on the product of spaces. (Throughout this paper all spaces will be assumed to be Hausdorff.)