On the rational homology of high dimensional analogues of spaces of long knots

By G. Arone, Victor Tourtchine. 2011

87 citations recorded by the source corpus.

We study high-dimensional analogues of spaces of long knots. These are spaces of compactly-supported embeddings (modulo immersions) of $\mathbb{R}^m$ into $\mathbb{R}^n$. We view the space of embeddings as the value of a certain functor at $\mathbb{R}^m$, and we apply manifold calculus to this functor. Our first result says that the Taylor tower of this functor can be expressed as the space of maps between infinitesimal bimodules over the little disks operad. We then show that the formality of the little disks operad has implications for the homological behavior of the Taylor tower. Our second result says that when $2m+1<n$, the singular chain complex of these spaces of embeddings is rationally equivalent to a direct sum of certain finite chain complexes, which we describe rather explicitly.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3ee4dec8a43ffe5be2ed7f7134019a1a3c97149a

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots (2011, background): “The structure of a right Γ-module is also intimately related to the commutative operad, more precisely we have seen in [40, 4] that a right Γ-module is the same thing as a weak bimodule over Com.”
The rational homotopy of mapping spaces of E${}_n$ operads (2017, methodology): “The map (27) (for m = n) induces a quasi-isomorphism KL ⊕ GC2n[1] ≃ → (HGC2n,n) L, where we consider the twisted hairy graph complex (HGC2n,n) L associated to the line graph (26).”
Spaces of smooth embeddings and configuration categories (2015, methodology): “It improves on some corollaries of related theorems in the work of Dwyer-Hess [13, 14], Arone-Turchin [3] and Turchin [35].”