We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in characteristic one over algebraic extensions of the infinite semifield of “max-plus integers” Zmax. An object of this category is a pair (E,K) of a semimodu...

We compute the A 1 -localization of several invariants schemes namely, topological Hochschild homology ( THH ), cyclic TC ) and periodic TP ). This procedure is quite brutal kills completed versions most these invariants. The main ingredient for vanishing statements de Rham cohomology (and, eventually, crystalline cohomology) in positive characteristics.

We extend the construction of Y-type invariants to null-homologous knots in rational homology three-spheres. By considering m -fold cyclic branched covers with a prime power, this extension provides new knot concordance S 3 . give computations some these for alternating and reprove independence results smooth group.

We conjecture an explicit formula for a cyclic analog of the Formality L∞-morphism [K]. We prove that its first Taylor component, the cyclic Hochschild-Kostant-Rosenberg map, is in fact a morphism (and a quasiisomorphism) of the complexes. To prove it we construct a cohomological version of the Connes-Tsygan bicomplex in cyclic homology. As an application of the cyclic Formality conjecture, we ...

Let G be a simple graph on n vertices, and let χG(λ) denote the chromatic polynomial of G. In this paper, we define the cyclic coloring complex, ∆(G), and determine the dimensions of its homology groups for simple graphs. In particular, we show that if G has r connected components, the dimension of (n− 3)rd homology group of ∆(G) is equal to (n− (r+ 1)) plus 1 r! |χG(0)|, where χG is the rth de...

In this article we investigate the autotopism group of the so-called cyclic semifield planes. We show that the group generated by the homology groups of the nuclei is already the full group of autotopisms that are linear with respect to the nuclei. The full autotopism group is also computed with the exception of one special subcase.