Volume 18, issue 3 (2014)

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

Gregory Arone and Victor Turchin

Geometry & Topology 18 (2014) 1261–1322

We study high-dimensional analogues of spaces of long knots. These are spaces of compactly supported embeddings (modulo immersions) of m into n. We view the space of embeddings as the value of a certain functor at 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.

embedding spaces, infinitesimal bimodules, formality
Mathematical Subject Classification 2010
Primary: 57R70
Secondary: 18D50, 18G55
Received: 29 March 2012
Revised: 14 October 2013
Accepted: 24 December 2013
Published: 7 July 2014
Proposed: Haynes Miller
Seconded: Bill Dwyer, Paul Goerss
Gregory Arone
Department of Mathematics
University of Virginia
Kerchof Hall
PO Box 400137
Charlottesville, VA 22904
Victor Turchin
Department of Mathematics
Kansas State University
138 Cardwell Hall
Manhattan, KS 66506