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248
The type of the classifying space for a family of subgroups
- J. Pure Appl. Algebra
"... 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 su ..."
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Cited by 105 (31 self)
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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.
Topology of symplectomorphism groups of rational ruled surfaces
- J. Amer. Math. Soc
"... cases M carries a family of symplectic forms ωλ, where λ> −1 determines the cohomology class [ωλ]. This paper calculates the rational (co)homology of the group Gλ of symplectomorphisms of (M, ωλ) as well as the rational homotopy type of its classifying space BGλ. It turns out that each group Gλ c ..."
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Cited by 62 (19 self)
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cases M carries a family of symplectic forms ωλ, where λ> −1 determines the cohomology class [ωλ]. This paper calculates the rational (co)homology of the group Gλ of symplectomorphisms of (M, ωλ) as well as the rational homotopy type of its classifying space BGλ. It turns out that each group Gλ contains a finite collection Kk, k = 0,..., ℓ = ℓ(λ), of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups “asymptotically commute”, i.e. all the higher Whitehead products that they generate vanish as λ → ∞. However, for each fixed λ there is essentially one nonvanishing product that gives rise to a “jumping generator” wλ in H ∗ (Gλ) and to a single relation in the rational cohomology ring H ∗ (BGλ). An analog of this generator wλ was also seen by Kronheimer in his study of families of symplectic forms on 4-manifolds using Seiberg–Witten theory. Our methods involve a close study of the space of ωλ-compatible almost complex structures on M. 1
Higher–order polynomial invariants of 3–manifolds giving lower bounds for the Thurston norm
, 2002
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On the topology of deformation spaces of Kleinian groups
"... Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(π1(M)) denote the space of (conjugacy classes of) discrete fathful representations of π1(M) into PSL2(C). The components of the interior MP(π1(M)) of AH(π1(M)) (as a subset of the appropriate representatio ..."
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Cited by 33 (13 self)
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Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(π1(M)) denote the space of (conjugacy classes of) discrete fathful representations of π1(M) into PSL2(C). The components of the interior MP(π1(M)) of AH(π1(M)) (as a subset of the appropriate representation variety) are enumerated by the space A(M) of marked homeomorphism types of oriented, compact, irreducible 3-manifold homotopy equivalent to M. In this paper, we give a topological enumeration of the components of the closure of MP(π1(M)) and hence a conjectural topological enumeration of the components of AH(π1(M)). We do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable Kleinian groups. We use this enumeration to exhibit manifolds M for which AH(π1(M)) has infinitely many components.
Hilbert modules and modules over finite von Neumann algebras and applications to L²-invariants
- MATH. ANN. 309, 247-285 (1997)
, 1997
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Noncommutative knot theory
, 2004
"... The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S 3 − K, considered as a module over the (commutative) Laurent polynomial ring, and the Blanchfield linking pairing defined on this module. From ..."
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Cited by 27 (10 self)
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The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S 3 − K, considered as a module over the (commutative) Laurent polynomial ring, and the Blanchfield linking pairing defined on this module. From the perspective of the knot group, G, these invariants reflect the structure of G (1) /G (2) as a module over G/G (1) (here G (n) is the n th term of the derived series of G). Hence any phenomenon associated to G (2) is invisible to abelian invariants. This paper begins the systematic study of invariants associated to solvable covering spaces of knot exteriors, in particular the study of what we call the n th higher-order Alexander module, G (n+1) /G (n+2) , considered as a Z[G/G (n+1)]–module. We show that these modules share almost all of the properties of the classical Alexander module. They are torsion modules with higher-order Alexander polynomials whose degrees give lower bounds for the knot genus. The modules have presentation matrices derived either from a group presentation or from a Seifert surface. They admit higher-order linking forms exhibiting self-duality. There are applications to estimating knot genus and to detecting fibered, prime and alternating knots. There are also surprising applications to detecting symplectic structures on 4–manifolds. These modules are similar to but different from those considered by the author, Kent Orr and Peter Teichner and are special cases of the modules considered subsequently by Shelly Harvey for arbitrary 3–manifolds.
A model structure on the category of pro-simplicial sets
- Trans. Amer. Math. Soc
"... Abstract. We study the category pro-SS of pro-simplicial sets, which arises in étale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SS so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict ..."
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Cited by 25 (5 self)
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Abstract. We study the category pro-SS of pro-simplicial sets, which arises in étale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SS so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for ind-spaces.
