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We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the G-CW-version EF(G) and the numerable G-space version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the Baum-Connes Conjecture about the topological K-theory of the reduced group C ∗-algebra, for the Farrell-Jones Conjecture about the algebraic K-and L-theory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.

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Riemann-Roch theorems assert that certain algebraically defined wrong way maps (transfers) in algebraic K–theory agree with topologically defined ones [BaDo]. Bismut and Lott [BiLo] proved such a Riemann–Roch theorem where the wrong way maps are induced by the projection of a smooth fiber bundle, and the topologically defined transfer map is the Becker–Gottlieb transfer. We generalize and refine their theorem, and prove a converse stating that the Riemann–Roch condition is equivalent to the existence of a fiberwise smooth structure. In the process, we prove a family index theorem where the K–theory used is algebraic K–theory, and the fiber bundles have topological (not necessarily smooth) manifolds as fibers.

We provide a "toolkit " of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories. We show how this toolkit can be used on quite different fields of applications. We demonstrate this with respect to 1. Bjorner's "Generalized Homotopy Complementation Formula" [4], 2. the topology of toric varieties, 3. the study of homotopy types of arrangements of subspaces, 4. the analysis of homotopy types of subgroup complexes.

This paper will show that, in the rank 2 case, the cohomology ring of this noncompact space is again generated by universal classes. A companion paper [23] gives a complete set of explicit relations between these generators

Abstract. We prove the Arnold conjecture for closed symplectic manifolds with π2(M) = 0 and cat M = dim M. Furthermore, we prove an analog of the Lusternik– Schnirelmann theorem for functions with “generalized hyperbolicity ” property.

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Abstract. Let M 0 µ denote S 2 ×S 2 endowed with a split symplectic form µσ⊕σ normalized so that µ ≥ 1 and σ(S 2) = 1. Given a symplectic embedding ι: Bc ֒ → M 0 µ of the standard ball of capacity c ∈ (0,1) into M 0 µ, consider the corresponding symplectic blow-up f M 0 µ,c. In this paper, we study the homotopy type of the symplectomorphism group Symp ( f M 0 µ,c) and that of the space ℑEmb(Bc, M 0 µ) of unparametrized symplectic embeddings of Bc into M 0 µ. Writing ℓ for the largest integer strictly smaller than µ, and λ ∈ (0,1] for the difference µ − ℓ, we show that the symplectomorphism group of a blow-up of “small ” capacity c < λ is homotopically equivalent to the stabilizer of a point in Symp(M 0 µ), while that of a blow-up of “large ” capacity c ≥ λ is homotopically equivalent to the stabilizer of a point in the symplectomorphism group of a nontrivial bundle CP 2 # CP 2 obtained by blowing down f M 0 µ,c. It follows that for c < λ, the space ℑEmb(Bc, M 0 µ) is homotopy equivalent to S 2 × S 2, while for c ≥ λ, it is not homotopy equivalent to any finite CW-complex. A similar result holds for symplectic ruled manifolds diffeomorphic to CP 2 # CP 2. By contrast, we show that the embedding spaces ℑEmb(Bc, CP 2) and ℑEmb(Bc1 ⊔Bc2, CP 2), if non empty, are always homotopy equivalent to the spaces of ordered configurations F(CP 2,1) ≃ CP 2 and F(CP 2,2). Our method relies on the theory of pseudo-holomorphic curves in 4-manifolds, on the computation of Gromov invariants in rational 4-manifolds, and on the inflation technique of Lalonde-McDuff. 1.

We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of Yetter’s Invariant by cohomology classes of crossed modules, defined