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Contribution to Collected Edition

Homotopy types and geometries below Spec Z


Manin,  Yuri I.
Max Planck Institute for Mathematics, Max Planck Society;

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Manin, Y. I., & Marcolli, M. (2020). Homotopy types and geometries below Spec Z. In P. Moree, A. Pohl, L. Snoha, & T. Ward (Eds.), Dynamics: topology and numbers (pp. 27-56). Providence: American Mathematical Society.

Cite as: http://hdl.handle.net/21.11116/0000-0006-50D6-E
After the first heuristic ideas about “the field of one element” F1 and “geometry in characteristics 1” (J. Tits, C. Deninger, M. Kapranov, A. Smirnov et al.), were developed several general approaches to the construction of “geometries below Spec Z”. Homotopy theory and the “the brave new algebra” were taking more and more important places in these developments, systematically explored by B. To¨en and M. Vaqui´e, among others. This article contains a brief survey and some new results on counting problems in this context, including various approaches to zeta–functions and generalised scissors congruences. We introduce a notion of F1 structures based on quasi-unipotent endomorphisms on homology. We also consider F1 structures based on the integral Bost–Connes algebra and its endomorphisms. In both cases we consider lifts of these structures, via an equivariant Euler charactetristic, to the level of Grothendieck rings and further lifts, via the formalism of assembler categories, to homotopy theoretic spectra.