Differentials on graph complexes

Differentials on Graph Complexes III - Deleting a Vertex

“…Under some weak conditions, it holds that D 2 = 0 and a grading can be settled such that D is a differential. We also obtain one further result about the spectral sequence of [2] for n = 0. The results will be used for the hairy graph complex in Section 7.…”

“…2.3. The spectral sequence of [2]. In the first paper of this series we introduced deformed differentials on the graph complexes above.…”

“…This paper is the third in the series of papers [2], [3], dealing with extra differentials on graph complexes. The basic idea is to deform the standard differential δ to δ + δ extra making the complex (almost) acyclic.…”

“…In the first paper of the series [2] the deformed differential was introduced for non-hairy graphs GC n for both parity of n. In the second paper of the series [3] the deformed differential was introduced for hairy graphs HGC m,n…”

“…However, in this paper we only get one small extra result in Corollary 3.9, strengthening the result from the first paper in this series about H (fGC n ) for even n (cf. [2,Corollary 4]). Computer calculations imply that the complex (GC −1 , D) is far from acyclic, but there is a hope that (GC −1 , δ + D) is acyclic.…”