Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

1-loop graphs and configuration space integral for embedding spaces

KEIICHI SAKAIa1 and TADAYUKI WATANABEa2

a1 Department of Mathematical Sciences, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan e-mail: ksakai@math.shinshu-u.ac.jp

a2 Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan e-mail: tadayuki@math.sci.hokudai.ac.jp

Abstract

We will construct differential forms on the embedding spaces Emb(j, n) for nj ≥ 2 using configuration space integral associated with 1-loop graphs, and show that some linear combinations of these forms are closed in some dimensions. There are other dimensions in which we can show the closedness if we replace Emb(j, n) by Emb(j, n), the homotopy fiber of the inclusion Emb(j, n) ↪ Imm(j, n). We also show that the closed forms obtained give rise to nontrivial cohomology classes, evaluating them on some cycles of Emb(j, n) and Emb(j, n). In particular we obtain nontrivial cohomology classes (for example, in H3(Emb(2, 5))) of higher degrees than those of the first nonvanishing homotopy groups.

(Received March 02 2010)

(Revised January 24 2011)

(Online publication January 10 2012)

Footnotes

† Partially supported by Grant-in-Aid for Young Scientists (B) 21740038, MEXT, Japan, Grant for Basic Science Research Projects, the Sumitomo Foundation and The Iwanami Fujukai Foundation.

‡ Partially supported by Grant-in-Aid for JSPS Fellows 08J01880, JSPS and Grants-in-Aid for Young Scientists (Start-up) 21840002, JSPS.