Given a compact manifold $M$ and a smooth map $g\colon M\to U(l\times l;\mathbb{C})$ from $M$ to the Lie group of unitary $l\times l$ matrices with entries in $\mathbb{C}$, we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized cyclic complex $\mathscr{N}_{\epsilon}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which is mapped to the odd Bismut–Chern character under the equivariant Chen integral map. It is also shown that the assignment $g\mapsto \mathrm{Ch}^-(g)$ induces a well-defined group homomorphism from the $K^{-1}$ theory of $M$ to the odd homology group of $\mathscr{N}_{\epsilon}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$.

中文翻译：

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全环同调和等变环空间同调中的奇特征类

给定一个紧凑的流形 $M$ 和一个光滑的映射 $g\colon M\to U(l\times l;\mathbb{C})$ 从 $M$ 到酉 $l\times l$ 矩阵的李群$\mathbb{C}$ 中的条目，我们构造了一个 Chern 字符 $\mathrm{Ch}^-(g)$，它位于等变（整个）循环 Chen 归一化循环复数 $\mathscr{N }_{\epsilon}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$ $M$，并且映射到等变陈积分映射下的奇数 Bismut-Chern 字符. 还表明，赋值 $g\mapsto \mathrm{Ch}^-(g)$ 从 $M$ 的 $K^{-1}$ 理论到奇同调群的定义明确的群同态$\mathscr{N}_{\epsilon}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$。