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I am working with non-compact Kac-Moody groups $\mathcal{K}$. We can use Milnor's join model for $E\mathcal{K}=\varinjlim \mathcal{K}^{*n}$, where $\mathcal{K}^{*n}$ is the iterated join (see page 20 …

Let $\mathcal{H}$ be a unitary universe for some group $G$. As $\mathcal{H}$ is a faithful representation the representation map is an injection $G \to \mathcal{U(H)}$, so there's a free $G$ action on …

Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I …

The formula is $$A \cdot B = \begin{bmatrix} A \otimes I & -I \otimes B^* \\ I \otimes B & A^* \otimes I \end{bmatrix}$$ the sign $- I \otimes B^*$ is to make associativity work out. I found it in …

So, if we have an infinite dimensional Hilbert space $H$ then the way you put a ring structure on $F(H)$ is by taking the isomorphism $H\oplus H \to H$ we can define the sum of two Fredholm operators …

I am trying to figure out some commutative diagrams and am having difficulty with this one. We have two fibrations of $B$, $E'$ and $E$ such that $E' \subset E$ and $E' \to E$ is a cofibration. $B$, $ …