Classifying spaces for commutativity of low-dimensional Lie groups

Camarena, Omar Antolín; Gritschacher, Simon; Villarreal, Bernardo

doi:10.1017/s0305004119000240

Cited by 11 publications

(19 citation statements)

References 12 publications

(104 reference statements)

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“…Now consider the inclusion k : BO(1) 2 → B com O(2). In [4], they show as well that k * (r) = 0, and since k * (w 1 ) = u + v, where u, v are the degree 1 generators of the polynomial algebra H * (BO (1)…”

Section: Transitionally Commutative Structures Over the Spherementioning

confidence: 87%

“…In [4], the authors show the existence of a class r ∈ H 2 (B com O(2); Z) satisfying j * (r) = 2e (5) where e ∈ H 2 (BSO(2); Z) is the Euler class of the universal oriented SO(2)bundle. The element g * 1 (e) ∈ H 2 (S 2 ; Z) is a generator, and yields an identification…”

Section: Transitionally Commutative Structures Over the Spherementioning

confidence: 99%

“…We claim that (φ 2 ) * (r) = 2r in H 2 (B com O(2); Z). In [4] it is shown that H 2 (B com O(2); Z) is generated by the classes r and W 1 , the latter being the image under ι * : H 2 (BO(2); Z) → H 2 (B com O(2); Z) of the Bockstein of w 1 ∈ H 1 (BO(2); F 2 ). Let us write (φ 2 ) * (r) = mr + ε 1 W 1 , with m ∈ Z and ε 1 ∈ {0, 1}.…”

Section: Transitionally Commutative Structures Over the Spherementioning

confidence: 99%

“…In [4] it is shown that π 2 (B com O(2)) ∼ = Z 3 . In addition to the TC structures f k : S 2 → B com O(2) on the trivial bundle, we also have the standard algebraic TC structures j • g m considered in Lemma 3.7.…”

Section: Transitionally Commutative Structures Over the Spherementioning

confidence: 99%

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Commutative cocycles and stable bundles over surfaces

Ramras

Villarreal

2019

Forum Mathematicum

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Commutative K-theory, a cohomology theory built from spaces of commuting matrices, has been explored in recent work of Adem, Gómez, Gritschacher, Lind, and Tillman. In this article, we use unstable methods to construct explicit representatives for the real commutative K-theory classes on surfaces. These classes arise from commutative O(2)-valued cocycles, and are analyzed via the point-wise inversion operation on commutative cocycles. 13 4.

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Section: Transitionally Commutative Structures Over the Spherementioning

confidence: 87%

Section: Transitionally Commutative Structures Over the Spherementioning

confidence: 99%

Section: Transitionally Commutative Structures Over the Spherementioning

confidence: 99%

Section: Transitionally Commutative Structures Over the Spherementioning

confidence: 99%

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Commutative cocycles and stable bundles over surfaces

Ramras

Villarreal

2019

Forum Mathematicum

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“…We also study two constructions that combine, in different ways, the spaces Hom(Z n , G) and Rep(Z n , G) for varying n. The space B com G, known as the classifying space for commutativity in G, is the geometric realization of the simplicial space Hom(Z • , G). Introduced by Adem-Cohen-Torres-Giese [2], this space has been studied by a variety of authors [3,4,5,27]. For instance, when G = U is the infinite unitary group, B com U represents commutative complex K-theory.…”

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confidence: 99%

Hilbert–Poincaré series for spaces of commuting elements in Lie groups

Ramras¹,

Stafa²

2018

Math. Z.

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In this paper we study homological stability for spaces Hom(Z n , G) of pairwise commuting n-tuples in a Lie group G. We prove that for each n 1, these spaces satisfy rational homological stability as G ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite dimensional analogues of these spaces, Comm(G) and BcomG, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In addition, we show that the rational homology of the space of unordered commuting n-tuples in a fixed group G stabilizes as n increases.Our proofs use the theory of representation stability -in particular, J. Wilson's theory of FI W -modules. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.1.1. Ordered commuting tuples. Let {G r } r 1 denote one of the classical families of compact, connected Lie groups -namely G r = SU(r), U(r), SO(2r + 1), Sp(r), or SO(2r) -or the complexifications thereof. We have standard inclusions G r ֒→ G r+1 (see Section 6).Theorem 1.1. Fix k, n 0 and let {G r } r 1 be one of the classical sequences of Lie groups listed above. Then the standard inclusions G r ֒→ G r+1 induce isomorphismsfor r k, andThese results are proved in Theorems 7.1 and 10.7, and Corollary 11.3 respectively. We prove similar results for G r -equivariant homology in Section 9. This result, and the others discussed below, also apply to the groups Spin(r) (Example 12.2), and to the projectivizations of the above sequences (Example 12.3), although for the projective groups there are no maps inducing the isomorphisms.

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Higher Generation by Abelian Subgroups in Lie Groups

Camarena

Gritschacher

Villarreal

2021

Transformation Groups

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To a compact Lie group G one can associate a space E(2;G) akin to the poset of cosets of abelian subgroups of a discrete group. The space E(2;G) was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and Gómez, and other authors. In this short note, we prove that G is abelian if and only if πi(E(2;G)) = 0 for i = 1; 2; 4. This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply connected if and only if the group is abelian.

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Classifying spaces for commutativity of low-dimensional Lie groups

Cited by 11 publications

References 12 publications

Commutative cocycles and stable bundles over surfaces

Commutative cocycles and stable bundles over surfaces

Hilbert–Poincaré series for spaces of commuting elements in Lie groups

Higher Generation by Abelian Subgroups in Lie Groups

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