The product formula in unitary deformation \(K\)-theory. (English) Zbl 1110.19004
For finitely generated groups \(G\) and \(H\), the author proves that there is a weak equivalence \({\mathcal{K}}G\wedge_{ku}{\mathcal{K}}H\simeq{\mathcal{K}}(G\times H)\) of \(ku\)-algebra spectra, where \(\mathcal{K}\) denotes the unitary deformation \(K\)-theory functor and \(ku\) the connective \(K\)-theory spectrum (Theorem 1). The author uses more than one weakly equivalent model for \({\mathcal{K}}G\). Suppose \({\mathcal{K}}G\) is the \(K\)-theory of a category of unitary representations of \(G\). Then one has a map of commutative \(ku\)-algebra spectra \({\mathcal{K}}G\wedge_{ku}{\mathcal{K}}H \to {\mathcal{K}}(G\times H)\) induced by the tensor product of representations. The result asserts that this map is a weak equivalence.
There is a filtration of \({\mathcal{K}}G\) by subspectra \({\mathcal{K}}G_n\). These subspectra correspond to representations of \(G\) whose irreducible components have dimension \(\leq n\). The first main step in the author’s proof consists in constructing a filtration of the spectrum \({\mathcal{K}}G\wedge_{ku}{\mathcal{K}}H\) that agrees with the existing filtration on \({\mathcal{K}}(G\times H)\). This is overcome by applying another approach to the model for \({\mathcal{K}}G\). Letting \(M_n\) denote the spectra which define a desired filtration of \({\mathcal{K}}G\wedge_{ku}{\mathcal{K}}H\), there exist maps of \(ku\)-algebras \(M_n \to {\mathcal{K}}(G\times H)_n\) which are induced from maps \({\mathcal{K}}G_p\wedge_{ku}{\mathcal{K}}H_q \to {\mathcal{K}}(G\times H)_n\) similar to the above one where \(pq \leq n\). The result has been established by showing that all these maps are weak equivalences.
Now, as pointed out, it seems that there is a remarkable similarity between the present product formula and the isomorphism \(R(G)\otimes R(H)\cong R(G\times H)\) of unitary representation rings. But it will be partly because both of them depend (except for a difference in degree) on the fact that every irreducible unitary representation of \(G\times H\) takes precisely the form of a tensor product of those of \(G\) and \(H\) (Lemma 37). In addition to this formula, making use of the above filtration of \({\mathcal{K}}G\), the author constructs spectral sequences for computing the homotopy groups of \({\mathcal{K}}G\) and \({\text{H}}{\mathbb{Z}}\wedge_{ku}{\mathcal{K}}G\) (Theorems 31 and 33).
There is a filtration of \({\mathcal{K}}G\) by subspectra \({\mathcal{K}}G_n\). These subspectra correspond to representations of \(G\) whose irreducible components have dimension \(\leq n\). The first main step in the author’s proof consists in constructing a filtration of the spectrum \({\mathcal{K}}G\wedge_{ku}{\mathcal{K}}H\) that agrees with the existing filtration on \({\mathcal{K}}(G\times H)\). This is overcome by applying another approach to the model for \({\mathcal{K}}G\). Letting \(M_n\) denote the spectra which define a desired filtration of \({\mathcal{K}}G\wedge_{ku}{\mathcal{K}}H\), there exist maps of \(ku\)-algebras \(M_n \to {\mathcal{K}}(G\times H)_n\) which are induced from maps \({\mathcal{K}}G_p\wedge_{ku}{\mathcal{K}}H_q \to {\mathcal{K}}(G\times H)_n\) similar to the above one where \(pq \leq n\). The result has been established by showing that all these maps are weak equivalences.
Now, as pointed out, it seems that there is a remarkable similarity between the present product formula and the isomorphism \(R(G)\otimes R(H)\cong R(G\times H)\) of unitary representation rings. But it will be partly because both of them depend (except for a difference in degree) on the fact that every irreducible unitary representation of \(G\times H\) takes precisely the form of a tensor product of those of \(G\) and \(H\) (Lemma 37). In addition to this formula, making use of the above filtration of \({\mathcal{K}}G\), the author constructs spectral sequences for computing the homotopy groups of \({\mathcal{K}}G\) and \({\text{H}}{\mathbb{Z}}\wedge_{ku}{\mathcal{K}}G\) (Theorems 31 and 33).
Reviewer: Haruo Minami (Nara)
MSC:
| 19L47 | Equivariant \(K\)-theory |
| 55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |
| 19L41 | Connective \(K\)-theory, cobordism |
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