The connective K-theory spectrum conjecture for surface groups

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Let MgM^g be a Riemann surface of genus gg, let π1M\pi_1 M be its fundamental group, and let Kdef(π1M)K^{\mathrm{def}}(\pi_1 M) denote the deformation KK-theory spectrum of this group. Let ku\mathbf{ku} be the connective KK-theory spectrum, and write Σ\Sigma for suspension and ∨\vee for the wedge sum. Spectrum decomposition conjecture. For any Riemann surface MgM^g, the spectrum Kdef(π1M)K^{\mathrm{def}}(\pi_1 M) is weakly equivalent, as a ku\mathbf{ku}-algebra, to

ku∨(⋁2gΣku)∨Σ2ku.\mathbf{ku}\vee\left(\bigvee_{2g}\Sigma\mathbf{ku}\right)\vee\Sigma^2\mathbf{ku}.

The conjecture proposes a spectrum-level refinement of the computation of the deformation KK-theory of surface groups; the proposed spectrum has the same homotopy groups as K∗def(π1M)K^{\mathrm{def}}_*(\pi_1 M). The compatibility of the spectrum-level construction with the ku\mathbf{ku}-algebra structure remains to be established.

References

Primary source

Daniel A. Ramras, “Yang-Mills theory over surfaces and the Atiyah-Segal theorem”, arXiv:0710.0681 (2018).

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