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Register a new userThe product structure of the equivariant K-theory of the based loop group of SU(2)
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Let G=SU(2) and let Omega G denote the space of continuous based loops in G, equipped with the pointwise conjugation action of G. It is a classical fact in topology that the ordinary cohomology H^*(Omega G) is a divided polynomial algebra Gamma[x]. The algebra Gamma[x] can be described as an inverse limit as k goes to infinity of the symmetric subalgebra in the exterior algebra Lambda(x_1, ...,x_k) in the variables x_1, ..., x_k. We compute the R(G)-algebra structure of the G-equivariant K-theory of Omega G in a way which naturally generalizes the classical computation of the ordinary cohomology ring of Omega G as a divided polynomial algebra Gamma[x]. Specifically, we prove that K^*_G(Omega G) is an inverse limit of the symmetric (S_{2r}-invariant) subalgebra of K^*_G((P^1)^{2r}), where the symmetric group S_{2r} acts in the natural way on the factors of the 2r-fold product (P^1)^{2r} and G acts diagonally via the standard action on each complex projective line P^1.
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Let $G=SU(2)$ and let $Omega G$ denote the space of based loops in SU(2). We explicitly compute the $R(G)$-module structure of the topological equivariant $K$-theory $K_G^*(Omega G)$ and in particular show that it is a direct product of copies of $K^*_G(pt) cong R(G)$. (We intend to describe in detail the $R(G)$-algebra (i.e. product) structure of $K^*_G(Omega G)$ in a forthcoming companion paper.) Our proof uses the geometric methods for analyzing loop spaces introduced by Pressley and Segal (and further developed by Mitchell). However, Pressley and Segal do not explicitly compute equivariant $K$-theory and we also need further analysis of the spaces involved since we work in the equivariant setting. With this in mind, we have taken this opportunity to expand on the original exposition of Pressley-Segal in the hope that in doing so, both our results and theirs would be made accessible to a wider audience.
For G a finite group and X a G-space on which a normal subgroup A acts trivially, we show that the G-equivariant K-theory of X decomposes as a direct sum of twisted equivariant K-theories of X parametrized by the orbits of the conjugation action of G on the irreducible representations of A. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of A to the subgroup of G which fixes the isomorphism class of the irreducible representation.
In this note we show that Waldhausens K-theory functor from Waldhausen categories to spaces has a universal property: It is the target of the universal global Euler characteristic, in other words, the additivization of the functor sending a Waldhausen category C to obj(C) . We also show that a large class of functors possesses such an additivization.
We show that the Waldhausen trace map $mathrm{Tr}_X colon A(X) to QX_+$, which defines a natural splitting map from the algebraic $K$-theory of spaces to stable homotopy, is natural up to emph{weak} homotopy with respect to transfer maps in algebraic $K$-theory and Becker-Gottlieb transfer maps respectively.
We define a $K$-theory for pointed right derivators and show that it agrees with Waldhausen $K$-theory in the case where the derivator arises from a good Waldhausen category. This $K$-theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator $K$-theory, as originally defined, is the best approximation to Waldhausen $K$-theory by a functor that is invariant under equivalences of derivators.
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