Cohomology of the space of commuting<i>n</i>–tuples in a compact Lie group

Baird, Thomas John

doi:10.2140/agt.2007.7.737

Cited by 40 publications

(121 citation statements)

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“…where we have used line (5) in the first equality and naturality of the K -theory product in the second. Up to the isomorphism in line (6) again, this is exactly the statement of the lemma.…”

Section: Definition 22mentioning

confidence: 99%

“…We also obtain some information about the K -theory and cohomology of the representation varieties Hom(Γ, U(n)), which have received a good deal of attention recently from a number of authors (see, for instance, [1,5] We were also motivated to put Lusztig's work in a more obviously representation theoretic-framework by Carlsson's deformation K -theory: in some sense, deformation K -theory develops related ideas in homotopy theory and algebraic K -theory. Carlsson associates to a group Γ a spectrum (in the sense of stable homotopy theory) K def (Γ), built from the (topological) category of finite dimensional unitary representations of Γ (see [25] for a description of the construction).…”

Section: Introductionmentioning

confidence: 99%

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A finite-dimensional approach to the strong Novikov conjecture

Ramras

Willett

2013

Algebr. Geom. Topol.

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A finite dimensional approach to the strong Novikov conjecture DANIEL RAMRAS RUFUS WILLETT GUOLIANG YUThe aim of this paper is to describe an approach to the (strong) Novikov conjecture based on continuous families of finite dimensional representations: this is partly inspired by ideas of Lusztig related to the Atiyah-Singer families index theorem, and partly by Carlsson's deformation K -theory. Using this approach, we give new proofs of the strong Novikov conjecture in several interesting cases, including crystallographic groups and surface groups. The method presented here is relatively accessible compared with other proofs of the Novikov conjecture, and also yields some information about the K -theory and cohomology of representation spaces.19K56, 55N15, 57R20; 20C99, 46L85, 46L80 IntroductionThe aim of this paper is to study the strong Novikov conjecture [19] for a finitely presented group Γ. If we assume that Γ has a finite classifying space BΓ, one version of this conjecture states that the analytic assembly mapis rationally injective; here the left hand side is the K -homology of BΓ and the right hand side is the K -theory of the maximal group C * -algebra of Γ. We give a definition of the analytic assembly map in Section 2 below. The strong Novikov conjecture implies the classical Novikov conjecture on homotopy invariance of higher signatures, as well as being closely related to several other famous conjectures. A naive version of our approach might proceed as follows. A finite dimensional unitary representationof Γ defines a vector bundle E ρ over BΓ via a well-known balanced product construction. E ρ defines an element [E ρ ] of the K -theory group K * (BΓ) and thus a 'detecting homomorphism'As is well-known 1 , ρ * factors through the analytic assembly map µ; hence µ(x) = 0 for any x ∈ K * (BΓ) such that there exists ρ : Γ → U(n) with ρ * (x) = 0. Thus if one can find 'enough' representations to detect all of K * (BΓ), one would have proved the strong Novikov conjecture.Unfortunately, this approach will not work: the bundles E ρ are flat, so Chern-Weil theory tells us that any 'detecting homomorphism' as in line (2) above is rationally trivial on reduced K -homology. One possible way to salvage the idea in the paragraph above is to use infinite dimensional representations. This led to the Fredholm representations of Miscenko [23], and subsequently to Kasparov's KK-theory [19]; both of these, and the closely related approach to the Novikov conjecture through the Baum-Connes conjecture [7] have proved enormously fruitful.In this paper we suggest a different approach. The central idea is not to use a single representation as in line (1) above, but instead a continuous family of representationsparametrized by a topological space X . Such a family defines a bundle E ρ over X × BΓ and thus a detecting homomorphismfrom the K -homology of BΓ to the K -theory of X via slant product with [E ρ ] ∈ K * (BΓ × X). This ρ * still factors through the analytic assembly map. The central result of this paper is that for...

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Section: Definition 22mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A finite-dimensional approach to the strong Novikov conjecture

Ramras

Willett

2013

Algebr. Geom. Topol.

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“…Let A ⊂ Z denote the subset of pairs (k 0 , k 1 ) such that k 0 and k 1 lie in some common maximal torus. The subspace A is preserved by the G-action, and the stabilizers of points in A all contain maximal tori of G. Applying ( [2], Theorem 3.3) we produce a G-equivariant cohomology isomorphism…”

Section: The Klein Bottlementioning

confidence: 99%

Moduli of flat SU(3)-bundles over a Klein bottle

Baird

2010

Mathematical Research Letters

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“…Theorem 1.1 is due to Torres-Giese and Sjerve [10] in the case that G is either SO(3), SU (2) or U (2). In their work, Torres-Giese and Sjerve determine the topological type of Hom(Z k , G) and compute its fundamental group via the Seifert-van Kampen Theorem.…”

Section: Introductionmentioning

confidence: 99%

“…Fix T a maximal torus in G, let N (T ) be the normalizer of T in G and W = N (T )/T the associated Weyl group. Following Baird [2], we consider the continuous surjection…”

Section: Introductionmentioning

confidence: 99%