Sinha constructed a cosimplicial space K N that gives a model for the space of long knots modulo immersions in R N , N 4. On the other hand, Lambrechts, Turchin and Volić showed that for N 4 the homology Bousfield-Kan spectral sequence associated to Sinha's cosimplicial space K N collapses at the E 2 page rationally. Their approach consists in first proving the formality of some other diagrams approximating K N and next deducing the collapsing result. In this paper, we prove directly the formality of Sinha's cosimplicial space, which immediately implies the collapsing result for N 3. Moreover, we prove that the isomorphism between the E 2 page and the homology of the space of long knots modulo immersions respects the Gerstenhaber algebra structure, when N 4.
We provide a complete understanding of the rational homology of the space of long links of m strands in R d , when d ≥ 4. First, we construct explicitly a cosimplicial chain complex, L• * , 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 L• * collapses at the E 2 page, that the homology Bousfield-Kan spectral sequence associated to the Munson-Volić cosimplicial model for the space of long links collapses at the E 2 page rationally, solving a conjecture of Munson-Volić. Our method enables us also to determine the rational homology of high dimensional analogues of spaces of long links. Our last result 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.