All Questions

Let $G$ be a compact connected Lie group. Recall that $\mathrm{H}^4(\mathrm{B}G;\mathbb{Z})$ is then a free abelian group of finite rank. Let us say that a class $c \in \mathrm{H}^4(\mathrm{B}G;\...

I'm reading the Book of John Roe, Elliptic Operators, Topology and Asymptotic Methods and got stuck at Lemma 2.27. i) How does this lemma show that a real vector bundle can be given by a pullback of ...

Given a group $$ G= PSU(N) \rtimes \mathbb{Z}_2, $$ where $PSU(N)$ is a projective special unitary group. Say $a \in PSU(N)$, $c \in \mathbb{Z}_2$, then $$ c a c= a^*, $$ which $c$ flips $a$ to its ...

Triality is a relationship among three vector spaces. It describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation ...

There is a canonical faithful orthogonal representation of the symmetric group $S_{n+1}$, for $n\geq 1$: $$ S_{n+1}\to O(n) $$ given as follows. (1). I regard $O(n)$ as the isometry group of the unit ...

I'm pretty confused about the precise relation of the integral and the real cohomology of the classifying space $BG$ of a compact Lie group $G$. The natural map $H^n(BG;\mathbb{Z})\to H^n(BG;\mathbb{R}...