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?

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

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

Applying this,

$K^{-n}(S^1\times X,\{0\}\times X) = K((S^1-\{0\})\times X\times \mathbb{R}^n)=K(X\times \mathbb{R}^{n+1})=K^{-(n+1)}(X)$

I just blindly applied the identity so I'm not completely sure.


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