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1-loop graphs and configuration space integral for embedding spaces

Published online by Cambridge University Press:  10 January 2012

KEIICHI SAKAI  and
TADAYUKI WATANABE
KEIICHI SAKAI
Affiliation:
Department of Mathematical Sciences, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan e-mail: [email protected]
TADAYUKI WATANABE
Affiliation:
Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan e-mail: [email protected]
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Abstract

We will construct differential forms on the embedding spaces Emb(j, n) for n − j ≥ 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 3 , May 2012 , pp. 497 - 533
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[ALV]Arone, G., Lambrechts, P. and Volić, I.Calculus of functors, operad formality, and rational homology of embedding spaces. Acta Math. 199 (2007), no. 2, 153–198.Google Scholar
[AT1]Arone, G. and Tourtchine, V. On the rational homology of high dimensional analogues of spaces of long knots, arXiv:1105.1576Google Scholar
[AT2]Arone, G. and Tourtchine, V. On the rational homotopy of high dimensional analogues of spaces of long knots, arXiv:1108.1001Google Scholar
[B1]Budney, R.Little cubes and long knots. Topology 46 (2007), 1–27.Google Scholar
[B2]Budney, R.A family of embedding spaces. Geom. Topol. Monogr. 13 (2008), 41–84.Google Scholar
[BT]Bott, R. and Taubes, C.On the self-linking of knots. J. Math. Phys. 35 (1994), 5247–5287.CrossRefGoogle Scholar
[CCL]Cattaneo, A., Cotta-Ramusino, P. and Longoni, R.Configuration spaces and Vassiliev classes in any dimensions. Algebr. Geom. Topol. 2 (2002), 949–1000.CrossRefGoogle Scholar
[CR]Cattaneo, A. and Rossi, C.Wilson surfaces and higher dimensional knot invariants. Comm. Math. Phys. 256 (2005), 513–537.CrossRefGoogle Scholar
[Ek]Ekholm, T.Differential 3-knots in 5-space with and without self-intersections. Topology 40 (2001), 157–196.CrossRefGoogle Scholar
[HKS]Habiro, K., Kanenobu, T. and Shima, A.Finite type invariants of ribbon 2-knots. Contemp. Math. 233 187–196.CrossRefGoogle Scholar
[HM]Hughes, F. and Melvin, P.The Smale invariant of a knot. Comment. Math. Helv. 60 (1985), 615–627.CrossRefGoogle Scholar
[K]Kosinski, A.Differential manifolds. Pure and Applied Mathematics 138 (Academic Press, Inc., Boston, MA, 1993).Google Scholar
[MM]Milnor, J. and Moore, J.On the structure of Hopf algebras. Ann. of Math. 81 (1965), 211–264.CrossRefGoogle Scholar
[MT]Mimura, M. and Toda, H.Topology of Lie groups. I, II. Transl. Math. Monogr. 91 (Amer. Math. Soc., Providence, RI, 1991).Google Scholar
[R]Rossi, C. Invariants of higher-dimensional knots and topological quantum field theories. PhD. Thesis. Zurich University (2002).Google Scholar
[Sa]Sakai, K.Configuration space integrals for embedding spaces and the Haefliger invariant. Journal of Knot Theory and Its Ramifications. 19 (12) (2010), 1597–1644.Google Scholar
[Si]Sinha, D.Operads and knot spaces. J. Amer. Math. Soc. 19 (2006), 461–486.Google Scholar
[Sm]Smale, S.The classification of immersions of spheres in Euclidean spaces. Ann. of Math. 69 (1959), 327–344.Google Scholar
[Th]Thurston, D. Integral Expressions for the Vassiliev Knot Invariants. Senior thesis. Harvard University (1995).Google Scholar
[To]Turchin, V.Hodge-type decomposition in the homology of long knots. J. Topol. 3 (2010), 487–534.Google Scholar
[Wa1]Watanabe, T.Configuration space integral for long n-knots and the Alexander polynomial, Algebr. Geom. Topol. 7 (2007), 47–92.CrossRefGoogle Scholar
[Wa2]Watanabe, T.On Kontsevich's characteristic classes for higher-dimensional sphere bundles. II. Higher classes. J. Topol. 2 (2009), 624–660.Google Scholar