# A K-theory isomorphism

I found this identities in a paper on Floquet topological classification which the author mentioned as a "well-known K-theory isomorphism"

$$K_{R}^{0,n}(S^1\times X, \{0\}\times X) = K_R^{0,n+1}(X).$$

Here $$S^1$$ is a circle (corresponding to time) and $$\{0\}$$ is a point in the circle (the initial time), $$X$$ is the Brillouin zone. The relative K-group implies that at $$t=0$$, the unitary must be identity for all $$k\in X$$. I initially thought this was just suspension but it didn't add up. Does anyone have any ideas?

• What is $K^{m,n}$? – Nihar Karve Jun 12 at 3:06

I found an identity in Karoubi (4.11 page 87) $$K^{-n}(X,Y):= K((X-Y)\times \mathbb{R}^n)$$
$$K^{-n}(S^1\times X,\{0\}\times X) = K((S^1-\{0\})\times X\times \mathbb{R}^n)=K(X\times \mathbb{R}^{n+1})=K^{-(n+1)}(X)$$