As is well known, whenever $X$ is a compacta space, the Atiyah-Janich theorem says that there is an identification $$[X,\mbox{Fred}(H) ]\cong K^0(X) $$ between the set of homotopy classes of maps from the space $X$ to the space of Fredholm operators on a (separable) Hilbert space and the $K^0$ group of $X$. My question is: How it looks the corresponding presentation for the $K^1$ group of $X$ in terms of Fredholm operators?
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7$\begingroup$ $$K^1(X) = \widetilde K(\Sigma X) = [\Sigma X, \text{Fred}_0] = [X, \Omega \text{Fred}_0].$$ It is a theorem (I believe of Atiyah and Singer) that $\Omega \text{Fred}_0$ is homotopy equivalent to the space of self-adjoint Fredholm operators on the same (complex) Hilbert space. $\endgroup$– mmeCommented Aug 2, 2019 at 17:58
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1$\begingroup$ If I remember correctly this is explained in the lecture notes of Dan Freed here web.ma.utexas.edu/users/dafr/M392C-2015/index.html $\endgroup$– Thomas RotCommented Aug 2, 2019 at 19:24
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