Little cubes and long knots
摘要
This paper gives a partial description of the homotopy type of K, the space of long knots in 3-dimensional Euclidean space. The primary result is the construction of a homotopy equivalence between K and the free little 2-cubes object over the space of prime knots. In proving the freeness result, a close correspondence is discovered between the Jaco-Shalen-Johannson decomposition of knot complements and the little cubes action on K. Beyond studying long knots in 3-space, we show that for any compact manifold M the space of embeddings of R^n x M in R^n x M with support in I^n x M admits an action of the operad of little (n+1)-cubes. If M=D^k this embedding space is the space of framed long n-knots in R^{n+k}, and the action of the little cubes operad is an enrichment of the monoid structure given by the connected-sum operation.
引用
@article{arxiv.math/0309427,
title = {Little cubes and long knots},
author = {Ryan Budney},
journal= {arXiv preprint arXiv:math/0309427},
year = {2007}
}
备注
35 pages, 17 figues. Revision history: V2 corrected mis-statement of a theorem of Hatcher's. V3 corrected statement and proof of the satellite decomposition of knots. V4 notational changes, extra figures, added some detail to proof of the freeness result. V5 formatting changes, references updated. v6 fixed the statement on when the space EC(j,D^n) is a j+1-fold loop space to take into account Haefliger's work