On the Second Homotopy Group of Spaces of Commuting Elements in Lie Groups
Research output: Contribution to journal › Journal article › Research › peer-review
Standard
On the Second Homotopy Group of Spaces of Commuting Elements in Lie Groups. / Adem, Alejandro; Gómez, José Manuel; Gritschacher, Simon.
In: International Mathematics Research Notices, Vol. 2022, No. 24, 2022, p. 19617–19689,.Research output: Contribution to journal › Journal article › Research › peer-review
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
TY - JOUR
T1 - On the Second Homotopy Group of Spaces of Commuting Elements in Lie Groups
AU - Adem, Alejandro
AU - Gómez, José Manuel
AU - Gritschacher, Simon
PY - 2022
Y1 - 2022
N2 - Let G be a compact connected Lie group and n⩾1 an integer. Consider the space of ordered commuting n-tuples in G, Hom(Zn,G), and its quotient under the adjoint action, Rep(Zn,G):=Hom(Zn,G)/G. In this article, we study and in many cases compute the homotopy groups π2(Hom(Zn,G)). For G simply connected and simple, we show that π2(Hom(Z2,G))≅Z and π2(Rep(Z2,G))≅Z and that on these groups the quotient map Hom(Z2,G)→Rep(Z2,G) induces multiplication by the Dynkin index of G. More generally, we show that if G is simple and Hom(Z2,G)\mathds1⊆Hom(Z2,G) is the path component of the trivial homomorphism, then H2(Hom(Z2,G)\mathds1;Z) is an extension of the Schur multiplier of π1(G)2 by Z. We apply our computations to prove that if BcomG\mathds1 is the classifying space for commutativity at the identity component, then π4(BcomG\mathds1)≅Z⊕Z, and we construct examples of non-trivial transitionally commutative structures on the trivial principal G-bundle over the sphere S4.
AB - Let G be a compact connected Lie group and n⩾1 an integer. Consider the space of ordered commuting n-tuples in G, Hom(Zn,G), and its quotient under the adjoint action, Rep(Zn,G):=Hom(Zn,G)/G. In this article, we study and in many cases compute the homotopy groups π2(Hom(Zn,G)). For G simply connected and simple, we show that π2(Hom(Z2,G))≅Z and π2(Rep(Z2,G))≅Z and that on these groups the quotient map Hom(Z2,G)→Rep(Z2,G) induces multiplication by the Dynkin index of G. More generally, we show that if G is simple and Hom(Z2,G)\mathds1⊆Hom(Z2,G) is the path component of the trivial homomorphism, then H2(Hom(Z2,G)\mathds1;Z) is an extension of the Schur multiplier of π1(G)2 by Z. We apply our computations to prove that if BcomG\mathds1 is the classifying space for commutativity at the identity component, then π4(BcomG\mathds1)≅Z⊕Z, and we construct examples of non-trivial transitionally commutative structures on the trivial principal G-bundle over the sphere S4.
U2 - 10.1093/imrn/rnab259
DO - 10.1093/imrn/rnab259
M3 - Journal article
VL - 2022
SP - 19617–19689,
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
SN - 1073-7928
IS - 24
ER -
ID: 282034292