Homotopy Invariance of K-theory.

Question

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, $K(\mathbb{R})\colon =K(S^1)=\mathbb Z$ and $K(\mathbb R^2)\colon=K(S^2)=\mathbb Z \oplus \mathbb Z$. But both the spaces are contractible. Am I making any mistake or $K$ ring is a homotopy invariant only if $X$ is (para)compact and Hausdorff?

Answer 1

If you extend $K$-theory to locally compact spaces via one-point compactification, you get induced homomorphisms only for proper maps. These are maps such that preimages of compact subsets are compact. No locally compact space can be contractible to a point via proper homotopies. However, $J = [0,1)$ is locally compact and has contractible one-point compactification. Thus all locally compact spaces which are properly homotopy equivalent to $J$ have trivial $K$-groups.