nLab framed little n-disk operad

Idea

The framed little $n$-disk operad is an operad in chain complexes that whose $n$-ary operations come from embedding $n$-dimensional disks in a larger one and rotating them. It is joint generalization of the little k-cubes operad and the framed little 2-disk operad.

Properties

Theorem

The shifted rational singular homology $H_{\bullet + d}(S X, \mathbb{Q})$ of the $n$-sphere space $S X = X^{S^n}$ of an oriented manifold $X$ of dimension $d$ naturally has the structure of a n-BV-algebra.

This appears as (CohenVoronov, theorem 5.3.3).

References

• Paolo Salvatore, Nathalie Wahl: Framed discs operads and Batalin-Vilkovisky algebras, The Quarterly Journal of Mathematics 54 2 (2003) 213–231 [pdf, doi:10.1093/qmath/hag012]

• Ralph Cohen, Alexander Voronov, section 5.3 of: Notes on string topology (arXiv:math/0503625)