arXiv subjects

Paul Arnaud Songhafouo Tsopméné

Publications and source records attributed to Paul Arnaud Songhafouo Tsopméné.

6 recordsLinked to original sources

arXiv2018

Formality of Sinha's cosimplicial model for long knots spaces and the Gerstenhaber algebra structure of homology

Sinha has constructed a cosimplicial space X for a fixed integer N. One of his main result states that for N > 3, X is a cosimplicial model for the space of long knots (modulo immersions). On the other hand, Lambrechts, Turchin and Volic showed that for N > 3 the homology Bousfield-Kan spectral sequence associated to Sinha's cosimplicial space collapses at the page 2 rationally. Their approach consists first to prove the formality of some other diagrams approximating X and next deduce the collapsing result. In this paper, we prove directly the formality of Sinha's cosimplicial space and this allows us to construct a very short proof of the collapsing result for N > 2. Moreover, we prove that the isomorphism between the page 2 and the rational homology of the space of long knots modulo immersions is of Poisson algebras, when N > 3.

math.AT↗

arXiv2018

Euler characteristics for spaces of string links and the modular envelope of $L_\infty$

We make calculations in graph homology which further understanding of the topology of spaces of string links, in particular calculating the Euler characteristics of finite-dimensional summands in their homology and homotopy. In doing so, we also determine the supercharacter of the symmetric group action on the positive arity components of the modular envelope of $L_\infty$.

math.AT↗

arXiv2017

Rational homology and homotopy of high dimensional string links

Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high dimensional anologues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher order Hochschild homology also called Hochschild-Pirashvili homology. In this paper, we generalize all these results to high dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under the study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of links spaces always when codimension is greater than two, i.e. always when the Goodwillie-Weiss calculus is applicable. Using Haefliger\rq{}s approach to calculate the groups of isotopy classes of higher dimensional links, we confirm our cojecture at the level of π_0.

math.AT↗

arXiv2017

Symmetric multiplicative formality of the Kontsevich operad

In his famous paper entitled "Operads and motives in deformation quantization", Maxim Kontsevich constructed (in order to prove the formality of the little d-disks operad) a topological operad, which is called in the literature the Kontsevich operad, and which is denoted Kd in this paper. This operad has a nice structure: it is a multiplicative symmetric operad, that is, it comes with a morphism from the symmetric associative operad. There are many results in the literature concerning the formality of Kd. It is well known (by Kontsevich) that Kd is formal over reals as a symmetric operad. It is also well known that Kd is formal as a multiplicative nonsymmetric operad. In this paper, we prove that the Kontsevich operad is formal over reals as a multiplicative symmetric operad, when d > 2.

math.AT↗

arXiv2017

The rational homology of spaces of long links

We provide a complete understanding of the rational homology of the space of long links of m strands in the euclidean space of dimension d > 3. First, we construct explicitly a cosimplicial chain complex whose totalization is quasi-isomorphic to the singular chain complex of the space of long links. Next we show (using the fact that the Bousfield-Kan spectral sequence associated to this cosimplcial chain complex collapses at the page 2) that the homology Bousfield-Kan spectral sequence associated to the Munson-Volic cosimplicial model for the space of long links collapses at the page 2 rationally, and this solves a conjecture of Munson-Volic. Our method enables us also to determine the rational homology of the high dimensional analogues of spaces of long links. The last result of this paper states that the radius of convergence of the Poincaré series for the space of long links (modulo immersions) tends to zero as m goes to the infinity.

math.AT↗

arXiv2014

McClure-Smith cosimplicial machinery and the cacti operad

McClure and Smith constructed a functor that sends a topological multiplicative operad O to an E_2 algebra TotO. They define in fact an operad D_2 (acting on the totalization TotO) weakly equivalent to the little 2-disks operad. On the other hand, Salvatore showed that D_2 is isomorphic to the cacti operad MS, which has a nice geometric description. He also built a geometric action of MS on TotO. In this paper we detail the McClure-Smith action and the cacti action. Our main result says that they are compatible in the sense that some squares must commute.

math.AT↗