On the structure of spaces of commuting elements in compact Lie groups

Ádem, Alejandro; Gómez, José Manuel

doi:10.1007/978-88-7642-431-1_1

Cited by 9 publications

(37 citation statements)

References 43 publications

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“…Their work was followed by work of T. Baird [9], who studied ordinary and equivariant cohomology, Baird-JeffreySelick [10],Ádem-Gómez [6,8,7],Ádem-Cohen-Gómez [3,4], Sjerve-Torres-Giese [27],Ádem-Cohen-Torres-Giese [5], Pettet-Suoto [26], Gómez-Pettet-Suoto [19], Okay [25]. Most of this work has been focused on the study of invariants such as cohomology, K-theory, connected components, homotopy type and stable decompositions.…”

Section: Summary Of Resultsmentioning

confidence: 99%

On Spaces of Commuting Elements in Lie Groups

Cohen¹,

Stafa²,

Reiner³

2014

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Abstract. The main purpose of this paper is to introduce a method to "stabilize" certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group G, namely Hom(Z n , G). We show that this stabilized space of homomorphisms decomposes after suspending once with "summands" which can be reassembled, in a sense to be made precise below, into the individual spaces Hom(Z n , G) after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilized space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilized space also allows the description of the additive reduced homology of the individual spaces Hom(Z n , G), with the order of the Weyl group inverted.

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Section: Summary Of Resultsmentioning

confidence: 99%

On Spaces of Commuting Elements in Lie Groups

Cohen¹,

Stafa²,

Reiner³

2014

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“…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%

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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“…Since e K is fixed by the conjugation action of K, the subgroup {e K } ∈ F is an isolated point in the given topology. Thus, for any ρ ∈ p −1 (e K ), the full connected component of ρ (which is path-connected) has trivial restriction to Γ (2) and we see that p −1 ({e K }) is the union of the connected components it intersects, completing the proof in this case.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 53%

“…For instance, if we take Γ to be the discrete Heisenberg group H 3 (Z), then Hom(Γ, SL 2 C) decomposes into a simply-connected component and a non simply-connected component. In fact, this phenomenon already occurs for Γ abelian as illustrated in Gómez-Adem [2] and Gómez-Pettet-Souto [15]. Theorem 1.5.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 80%