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Odd characteristic classes in entire cyclic homology and equivariant loop space homology. (English) Zbl 1494.55013

Motivated by the odd dimensional variant of Atiyah-Singer’s ‘index’ theorem, for \(M\) a compact manifold (possibly with boundary) and a smooth map \(g: M\to U(l\times l;\mathbb{C})\) from \(M\) to the Lie group of unitary \(l\times l\) matrices with entries in \(\mathbb{C}\), the authors construct a canonical element \(Ch^-(g)\in\mathcal{N} _\epsilon^-(\Omega_\mathbb{T} (M\times\mathbb{T}))\) in the odd part of the (reduced) Chen-normalized entire cyclic complex of \(\Omega_\mathbb{T} (M\times\mathbb{T})\), the smooth \(\mathbb{T}\)-invariant differential forms on \(M\times\mathbb{T}\) as a locally convex unital DGA.
This is suggested by the program proposed by Getzler, Jones and Petrack for infinite dimensional localization, where the idea is to take as model for \(\Omega(LM)\) the space of equivariant Chen integrals, and in particular, one can have an equivariant Chen integral map \(\rho:\mathcal{N}_\epsilon(\Omega_\mathbb{T}(M\times\mathbb{T}))\to\widehat{\Omega}(LM)\) to a completed space of smooth differential forms on the smooth loop space \(LM\). Note that the consideration of the loop space \(LM\) was motivated earlier by Atiyah and Bismut for the even-dimensional case. The authors show that if \(M\) is closed then the element \(Ch^-(g)\) they construct satisfies the three properties required to obtain a proof of the odd dimensional variant of Atiyah-Singer’s ‘index’ theorem within the GJP-program for infinite dimensional localization.
The authors also show that the assignment \(g\mapsto Ch^-(g)\) induces a well-defined group homomorphism \(K^{-1}(M)\to\mathcal{N} (\Omega_\mathbb{T} (M\times\mathbb{T}))\) from the \(K^{-1}\)-group of \(M\) to a quotient of the space of cyclic chains of the DGA \(\Omega_\mathbb{T} (M\times\mathbb{T})\). Finally, with the even variant of \(Ch^-(g)\) having been previously defined, the authors establish an even/odd periodicity, relating these two constructions.

MSC:

55N91 Equivariant homology and cohomology in algebraic topology
19D55 \(K\)-theory and homology; cyclic homology and cohomology
55P35 Loop spaces
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