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24
Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
Operads and Chain Rules for the Calculus of Functors
"... Abstract. We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain ..."
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Cited by 11 (0 self)
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Abstract. We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct new models for the Goodwillie derivatives of functors of spectra. These models allow for natural composition maps that yield operad and module structures. Then, we use a cosimplicial cobar construction to transfer this structure to functors of topological spaces. A form of Koszul duality for operads of spectra plays a key role in this. In a landmark series of papers, [16], [17] and [18], Goodwillie outlines his ‘calculus of homotopy functors’. Let F: C → D (where C and D are each either Top ∗, the category of pointed topological spaces, or Spec, the category of spectra) be a pointed homotopy functor. One of the things that Goodwillie does is associate with F a sequence of spectra, which are called the derivatives of F.
UMKEHR MAPS
, 711
"... Abstract. In this note we study umkehr maps in generalized (co)homology theories arising from the PontrjaginThom construction, from integrating along fibers, pushforward homomorphisms, and other similar constructions. We consider the basic properties of these constructions and develop axioms which ..."
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Cited by 5 (1 self)
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Abstract. In this note we study umkehr maps in generalized (co)homology theories arising from the PontrjaginThom construction, from integrating along fibers, pushforward homomorphisms, and other similar constructions. We consider the basic properties of these constructions and develop axioms which any umkehr homomorphism must satisfy. We use a version of Brown representability to show that these axioms completely characterize these homomorphisms, and a resulting uniqueness theorem follows. Finally, motivated by constructions in string topology, we extend this axiomatic treatment of umkehr homomorphisms to a fiberwise setting. 1.
Fixed point theory and trace for bicategories
, 2007
"... The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point inde ..."
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Cited by 4 (1 self)
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The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is the identification of these different invariants. We define a generalization of the trace in symmetric monoidal categories to a trace in bicategories with shadows. We show the invariants used in the converse of the Lefschetz fixed point theorem are examples of this trace and that the functoriality of the trace provides some of the necessary identifications. The methods used here do not use simplicial techniques and so generalize readily to other contexts. iii Contents
Characteristic cohomotopy classes for families of 4manifolds
 Forum Math
"... ABSTRACT. Families of smooth closed oriented 4manifolds with a complex spin structure are studied by means of a family version of the BauerFuruta invariants. The definition is given in the context of parametrised stable homotopy theory, but an interpretation in terms of characteristic cohomotopy c ..."
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Cited by 3 (3 self)
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ABSTRACT. Families of smooth closed oriented 4manifolds with a complex spin structure are studied by means of a family version of the BauerFuruta invariants. The definition is given in the context of parametrised stable homotopy theory, but an interpretation in terms of characteristic cohomotopy classes on Thom spectra associated to the classifying spaces of complex spin diffeomorphism groups is given as well. The theory is illustrated with families of K3 surfaces and mapping tori of diffeomorphisms. It is also related to equivariant invariants.
On the construction of functorial factorizations for model categories
, 2012
"... Abstract. We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use “algebraic ” characterizations of fibrations to produce factorizations that have the desired lifting properties in a com ..."
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Cited by 3 (2 self)
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Abstract. We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use “algebraic ” characterizations of fibrations to produce factorizations that have the desired lifting properties in a completely categorical fashion. We illustrate these methods in the case of categories enriched, tensored, and cotensored in spaces, proving the existence of Hurewicztype model structures, thereby correcting an error in earlier attempts by others. Examples include the categories of (based) spaces, (based) Gspaces, and diagram spectra among others. 1.
DIAGRAM SPACES, DIAGRAM SPECTRA, AND SPECTRA OF UNITS
, 908
"... Abstract. We compare the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM Smodules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor Ω ∞. We prove that these models for sp ..."
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Cited by 2 (0 self)
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Abstract. We compare the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM Smodules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor Ω ∞. We prove that these models for spaces are Quillen equivalent and that the infinite loop space functors Ω ∞ agree. This comparison is then used to show that two different constructions of the spectrum of units gl1R of a structured ring spectrum R agree. Contents
Model structures on the category of small double categories, Algebraic and Geometric Topology 8
, 2008
"... Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak ..."
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Cited by 2 (2 self)
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Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves
The refined transfer, bundle structures and algebraic ktheory
, 2007
"... We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits a reduction to a fiber bundle with compact topological manifold fibers. The criteria lead to an unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjectiv ..."
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We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits a reduction to a fiber bundle with compact topological manifold fibers. The criteria lead to an unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjective splitting of the assembly map for Waldhausen’s functor A(X). We also give concrete examples of fibrations having a reduction to a fiber bundle with compact topological manifold fibers but which fail to admit a compact fiber smoothing. The examples are detected by algebraic Ktheory invariants. We consider a refinement of the BeckerGottlieb transfer. We show that a version of the axioms described by Becker and Schultz uniquely determines the refined transfer for the class of fibrations admitting a reduction to a fiber bundle with compact topological manifold fibers. In an
THOMOTOPY AND REFINEMENT OF OBSERVATION (V) : STRØM MODEL STRUCTURE FOR BRANCHING AND MERGING
, 2005
"... Abstract. We check that there exists a model structure on the category of flows whose weak equivalences are the Shomotopy equivalences. As an application, we prove that the generalized Thomotopy equivalences preserve the branching and merging homology theories of a flow. The method of proof is com ..."
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Cited by 1 (1 self)
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Abstract. We check that there exists a model structure on the category of flows whose weak equivalences are the Shomotopy equivalences. As an application, we prove that the generalized Thomotopy equivalences preserve the branching and merging homology theories of a flow. The method of proof is completely different from the one of the third part of this series of papers. Contents