# Questions tagged [topological-k-theory]

Topological K-theory is a generalized cohomology theory, for which $K_0(X)$ is the Grothendieck group of isomorphism classes of vector bundles over topological space $X$. See also (algebraic-k-theory).

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### complexification of real vector bundles on $S^4$ and $S^8$

Prove there is a surjection $\tilde{KO}(S^8) \to \tilde{K}(S^8)$.(similarly for $S^4$) My idea: It is equivalent to show in every class of complex vector bundles on $S^8(S^4)$, there is a ...
1 vote
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### K Group Determined by Chern Classes

Why is it that the complex reduced K group of $\mathbb{CP}^2$ is determined by Chern classes $c_1$ and $c_2$? I am aware of the fact that the cohomology ring of complex Grassmannians is generated by ...
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### K functor of riemann surface

I have read about K-theory, but i didnt find any computational ideas there. Suppose i have a riemann surface of genus g - $S_g$, how can i compute $K_0(S_g)$?
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### Bundle homomorphisms and sections of Hom-bundle

I reading "Complex Topological K-theory" by Efton Park and came across exercise 1.4. Let $V,W$ be vector bundles over a compact Hausdorff space $X$. a) Show that the collection $Hom(V,W)$ of ...
1 vote
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### Lemma 3.6.9 from Complex Topological K-Theory by Efton Park

The lemma goes like this (in author's notation, $\pi: V\to X$ is the projection map; if $f: Y\to X$, and $W$ is a vector bundle over $X$, then $f^* W$ is the pullback of $W$ over $Y$): Let $X$ be a ...
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### What is the $\langle \cdot, \cdot \rangle$ operator when calculating Stiefel-Whitney numbers?

I've been trying to understand Stiefel-Whitney numbers and I'm reading this paper. On page 9, the author defines the Stiefel-Whitney number using an operator $\langle \cdot, \cdot \rangle$. I've tried ...
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### Homotopy Invariance of K-theory.

For a locally compact Hausdorff space $X$, the $K$ ring is defined to be the $K$ ring of its one-point compactification, i.e. $K(X)\colon =K(X^+)$. Therefore, $K(\mathbb{R})\colon =K(S^1)=\mathbb Z$ ...
1 vote
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### Calculating the equivariant K-group $K_G^*(S^1)$ - where's the mistake?

Let a finite group $G$ act on the circle $S^1$ via a group homomorphism $\varphi \colon G \to S^1$. Let $K = \ker \varphi$. I wish to calculate the equivariant K-theory group $K_G^*(S^1)$. One method ...
1 vote
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### Associated fibre bundle to a $(\Gamma,\alpha)$-equivariant $G$-principal bundle

Let $\Gamma$ and $G$ be compact Lie groups and $\alpha:\Gamma\to Aut(G)$ group homomorphism with the condition that $(\gamma,g)\mapsto \alpha(\gamma)(g)$ is continuous. $(\Gamma,\alpha,G)$-bundles are ...
1 vote
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### Collapsing of Kunneth formula for equivariant K-theory of homogeneous spaces.

Minami in "K-groups of symmetric spaces" (equations 1.1, 1.2) states the following, originally due to Hodgkins: Suppose that $G$ is a compact connected Lie group such that $\pi_1(G)$ is ...
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### Hatcher K-Theory continuity of $\alpha \to \inf_{(x,z)\in X \times S^1} \det|\alpha(x,z)|$

Does Hatcher makes an error at page $45$ afirming that $f:\alpha \mapsto \inf_{(x,z)\in X \times S^1} \det|\alpha(x,z)|$ is continuous? I didn't find any reference of continuity of $\inf$ of ...
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