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We know that there is a map from $h:\pi_{i}^{st}(pt)\rightarrow KO_{i}(pt)$ and we know all the $KO_{i}(pt)$ by Bott periodicy: they are $Z, Z_{2},Z_{2},0,Z,0,0,0$. We also know $\pi_{i}^{st}(pt)$ for …

Let X be a space with $Z/2$ action. There is a map from $K(X)$ to the equivariant K-group $K_{Z_{2}}(X)$, which is called "the induction map". (It is a standard operation in equivariant stable homotop …

Let $G_{1}, G_{2}, \cdots$ be a countably infinite sequence of finite groups. It is well-known that the group homology $H_{n}(BG_{i};\mathbb{Q})=0$ for any $n\geq 1$. Let $X=\prod^{\infty}_{i=1}BG_{i} …

Let $X$ be a projective variety over $\mathbb{C}$. Is there a way to define some number $\tilde{\chi}(X)\in \mathbb{Z}$ satisfying both of the following two properties? $\boldsymbol{(1)} \;$ When $X$ …