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I'm familiar with the definition of Chern character for a vector bundle. This leads to the definition of Chern character for $K$ theory (even theory) with values in even cohomology (the definition involves Chern classes). How the Chern character for the odd $K$-tehory is defined? It should land in odd cohomology theory, so I don't know what is the natural candidate.

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I think it is something like this: Recall that $K^{-1}(X):=\overline{K}(S(X_+))\cong\overline {K^0}(S(X))\oplus \overline{K^0}(S^1)\cong \overline {K^0}(S(X))$. Hence we can define a chern character (coefficients of cohomology are with rational coefficients) $$ K^{-1}(X)\otimes\mathbb{Q}\cong\overline{K^0}(SX)\otimes\mathbb{Q}\rightarrow \overline{H^{\mathrm{ev}}}(SX). $$ Using long exact sequences in cohomology, realize that $\overline{H^{\mathrm{ev}}}(SX)=H^{\mathrm{odd}}(X)$. Compose the chern character above with the map $\overline{H^{\mathrm{ev}}}(SX)\rightarrow H^{\mathrm{odd}}(X)$, to obtain a map $K^{-1}(X)\otimes\mathbb{Q}\rightarrow H^\mathrm{odd}(X)$.

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