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1-loop graphs and configuration space integral for embedding spaces


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.

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[ALV] G. Arone , P. Lambrechts and I. Volić Calculus of functors, operad formality, and rational homology of embedding spaces. Acta Math. 199 (2007), no. 2, 153198.

[B1] R. Budney Little cubes and long knots. Topology 46 (2007), 127.

[BT] R. Bott and C. Taubes On the self-linking of knots. J. Math. Phys. 35 (1994), 52475287.

[CCL] A. Cattaneo , P. Cotta-Ramusino and R. Longoni Configuration spaces and Vassiliev classes in any dimensions. Algebr. Geom. Topol. 2 (2002), 9491000.

[CR] A. Cattaneo and C. Rossi Wilson surfaces and higher dimensional knot invariants. Comm. Math. Phys. 256 (2005), 513537.

[Ek] T. Ekholm Differential 3-knots in 5-space with and without self-intersections. Topology 40 (2001), 157196.

[HKS] K. Habiro , T. Kanenobu and A. Shima Finite type invariants of ribbon 2-knots. Contemp. Math. 233 187196.

[HM] F. Hughes and P. Melvin The Smale invariant of a knot. Comment. Math. Helv. 60 (1985), 615627.

[MM] J. Milnor and J. Moore On the structure of Hopf algebras. Ann. of Math. 81 (1965), 211264.

[Sa] K. Sakai Configuration space integrals for embedding spaces and the Haefliger invariant. Journal of Knot Theory and Its Ramifications. 19 (12) (2010), 15971644.

[Si] D. Sinha Operads and knot spaces. J. Amer. Math. Soc. 19 (2006), 461486.

[Sm] S. Smale The classification of immersions of spheres in Euclidean spaces. Ann. of Math. 69 (1959), 327344.

[To] V. Turchin Hodge-type decomposition in the homology of long knots. J. Topol. 3 (2010), 487534.

[Wa1] T. Watanabe Configuration space integral for long n-knots and the Alexander polynomial, Algebr. Geom. Topol. 7 (2007), 4792.

[Wa2] T. Watanabe On Kontsevich's characteristic classes for higher-dimensional sphere bundles. II. Higher classes. J. Topol. 2 (2009), 624660.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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