### CHARACTERISTIC CLASSES IN BOREL COHOMOLOGY J.P.

, 1985

"... Let G be a topological group and let EC be a free contractible G-space. The Bore1 construction on a G-space X is the orbit space Xo = EC x G X. When asked what equivariant cohomology is, most people would answer Bore1 cohomology, namely H:(X) = H*(X,). This theory has the claim of priority and the ..."

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Let G be a topological group and let EC be a free contractible G-space. The Bore1 construction on a G-space X is the orbit space Xo = EC x G X. When asked what equivariant cohomology is, most people would answer Bore1 cohomology, namely H:(X) = H*(X,). This theory has the claim of priority and the merit of ready com-putability, and many very beautiful results have been proven with it. However, it suffers from the defects of its virtues. Precisely, it is ‘invariant’, in the sense that a G-map f: X + Y which is a nonequivariant homotopy equivalence induces an isomorphism on Bore1 cohomology. A quick way to see this is to observe that 1 x f: EC xX+ EC x Y is a map of principal G-bundles with base map lxof=fo:Xo-‘Y,, so that fc is a weak homotopy equivalence. As explained in [l], this invariance property is the crudest of a hierarchy of such properties that a theory might have. We shall show how to compute all characteristic classes in any invariant equivariant cohomology theory, the conclusion being that no such theory is powerful enough to support a very useful theory of characteristic classes. As ex-plained in [2], a less crude invariance property can sometimes be exploited to obtain

### Contents

"... 3. Coherent families of connected covers 7 4. Orientability of spherical G-fibrations 8 ..."

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3. Coherent families of connected covers 7 4. Orientability of spherical G-fibrations 8

### 9 1990 Kluwer Academic Publishers. Printed in the Netherlands. An Equivariant Novikov Conjecture Dedicated to Alexander Grothendieck

, 1989

"... Abstract. We discuss and formulate the correct equivariant generalization of the strong Novikov conjecture. This will be the statement that certain G-equivariant higher signatures (living in suitable equivariant K-groups) are invariant under G-maps of manifolds which, nonequivariantly, are homotopy ..."

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Abstract. We discuss and formulate the correct equivariant generalization of the strong Novikov conjecture. This will be the statement that certain G-equivariant higher signatures (living in suitable equivariant K-groups) are invariant under G-maps of manifolds which, nonequivariantly, are homotopy equivalences preserving orientation. We prove this conjecture for manifolds modeled on a complete Riemannian manifold of nonpositive curvature on which G (a compact Lie group) acts by isometries. We also use the theory of harmonic maps to construct (in some cases) G-maps into such model spaces. Key words. Novikov conjecture, equivafiant K-theory, G-pseudoequivalence, C*-algebra, fundamental groupoid, KK-theory 1. Formulation of the Problem and of the Main Results

### Elmendorf’s Theorem via Model Categories

, 2010

"... relates the equivariant homotopy theory of G-spaces to a homotopy theory of diagrams using fixed point sets. The diagrams are indexed by a topological category OG with objects the orbit spaces {G/H}H for the closed subgroups ..."

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relates the equivariant homotopy theory of G-spaces to a homotopy theory of diagrams using fixed point sets. The diagrams are indexed by a topological category OG with objects the orbit spaces {G/H}H for the closed subgroups

### EQUIVARIANT PHANTOM MAPS

, 2001

"... Abstract. A successful generalization of phantom map theory to the equivariant case for all compact Lie groups is obtained in this paper. One of the key observations is the discovery of the fact that homotopy fiber of equivariant completion splits as product of equivariant Eilenberg-Maclane spaces w ..."

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Abstract. A successful generalization of phantom map theory to the equivariant case for all compact Lie groups is obtained in this paper. One of the key observations is the discovery of the fact that homotopy fiber of equivariant completion splits as product of equivariant Eilenberg-Maclane spaces which seems impossible at first sight by the example of Triantafillou[19]. 1.

### MULTICURVES AND EQUIVARIANT COHOMOLOGY

, 2008

"... Abstract. Let A be a finite abelian group. We set up an algebraic framework for studying A-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal groups. We compute the equivariant cohomology of many spaces in these terms, including projective bundles (a ..."

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Abstract. Let A be a finite abelian group. We set up an algebraic framework for studying A-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal groups. We compute the equivariant cohomology of many spaces in these terms, including projective bundles (and associated Gysin maps), Thom spaces, and infinite Grassmannians. 1.

### The

, 1998

"... Spaces of maps into classifying spaces for equivariant crossed complexes, II: ..."

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Spaces of maps into classifying spaces for equivariant crossed complexes, II:

### EQUIVARIANT CELLULAR HOMOLOGY AND ITS APPLICATIONS

, 2001

"... Abstract. In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincaré formula and an equivariant Lefschetz theorem. ..."

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Abstract. In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincaré formula and an equivariant Lefschetz theorem.

### CONTINUOUS FUNCTORS AS A MODEL FOR THE EQUIVARIANT STABLE HOMOTOPY CATEGORY

, 2005

"... Abstract. In this paper, we investigate the properties of the category of equivariant diagram spectra indexed on the category WG of based G-spaces homeomorphic to finite G-CW-complexes for a compact Lie group G. Using the machinery of [10], we show that there is a “stable model structure ” on this c ..."

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Abstract. In this paper, we investigate the properties of the category of equivariant diagram spectra indexed on the category WG of based G-spaces homeomorphic to finite G-CW-complexes for a compact Lie group G. Using the machinery of [10], we show that there is a “stable model structure ” on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal G-spectra. We construct a second “absolute stable model structure ” which is Quillen equivalent to the “stable model structure”. Our main result is a concrete identification of the fibrant objects in the absolute stable model structure. There is a model-theoretic identification of the fibrant continuous functors in the absolute stable model structure as functors Z such that for A ∈ WG the collection {Z(A ∧ S W)} form an Ω-G-prespectrum as W varies over the universe U. We show that a functor is fibrant if and only if it takes G-homotopy pushouts to G-homotopy pullbacks and is suitably compatible with equivariant Atiyah duality for orbit spaces G/H+ which embed in U. Our motivation for this work is the development of a recognition principle for equivariant infinite loop spaces. The description of fibrant objects in the absolute stable model structure makes it clear that in the equivariant setting we cannot hope for a comparison between the category of equivariant continuous functors and equivariant Γ-spaces, except when G is finite. We provide an explicit analysis of the failure of