Poisson structures on the homology of the space of knots

In this article we study the Poisson algebra structure on the homology of the totalization of a fibrant cosimplicial space associated with an operad with multiplication. This structure is given as the Browder operation induced by the action of little disks operad, which was found by McClure and Smith. We show that the Browder operation coincides with the Gerstenhaber bracket on the Hochschild homology, which appears as the E2-term of the homology spectral sequence constructed by Bousfield. In particular we consider a variant of the space of long knots in higher dimensional Euclidean space, and show that Sinha's homology spectral sequence computes the Poisson algebra structure of the homology of the space. The Browder operation produces a homology class which does not directly correspond to chord diagrams.Comment: This is the version published by Geometry & Topology Monographs on 19 March 200

Deloopings of the spaces of long embeddings

The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give a delooping of the long embedding space, which can be regarded as a version of Morlet-Burghelea-Lashof's delooping of the diffeomorphism group of the disk relative to the boundary. As a corollary, we show that the homotopy fiber is weakly equivalent to a space on which the framed little disks operad acts possibly nontrivially, and hence its rational homology is a (higher) BV-algebra in a stable range of dimensions.Comment: 8 pages, title changed, to appear in Fundamenta Mathematica

Configuration space integrals for embedding spaces and the Haefliger invariant

Let K be the space of long j-knots in R^n. In this paper we introduce a graph complex D and a linear map I from D to the de Rham complex of K via configuration space integral, and prove that (1) when both n>j>=3 are odd, the map I is a cochain map if restricted to graphs with at most one loop component, (2) when n-j>=2 is even, the map I is a cochain map if restricted to tree graphs, and (3) when n-j >=3 is odd, the map I added a correction term produces a (2n-3j-3)-cocycle of K which gives a new formulation of the Haefliger invariant when n=6k, j=4k-1 for some k.Comment: 41 pages, many figures (v2: Theorem 1.3 and its proof have been improved. many minor corrections. v3: Theorem 1.3 and its proof have been revised, since Lemma 5.26 in v2 was wrong. v4: The proof of Theorem 1.3 has been fully revised. To appear in J. Knot Theory Ramifications

1-loop graphs and configuration space integral for embedding spaces

We will construct differential forms on the embedding spaces Emb(R^j,R^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(R^j,R^n) by fEmb(R^j,R^n), the homotopy fiber of the inclusion Emb(R^j,R^n) -> Imm(R^j,R^n). We also show that the closed forms obtained give rise to nontrivial cohomology classes, evaluating them on some cycles of Emb(R^j,R^n) and fEmb(R^j,R^n). In particular we obtain nontrivial cohomology classes (for example, in H^3(Emb(R^2,R^5))) of higher degrees than those of the first nonvanishing homotopy groups.Comment: 35 pages, to appear in Mathematical Proceedings of the Cambridge Philosophical Societ

An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3

Our main object of study is a certain degree-one cohomology class of the space K of long knots in R^3. We describe this class in terms of graphs and configuration space integrals, showing the vanishing of some anomalous obstructions. To show that this class is not zero, we integrate it over a cycle studied by Gramain. As a corollary, we establish a relation between this class and (R-valued) Casson's knot invariant. These are R-versions of the results which were previously proved by Teiblyum, Turchin and Vassiliev over Z/2 in a different way from ours.Comment: 11 pages, 4 figure

The space of short ropes and the classifying space of the space of long knots

ArticleAlgebraic & Geometric Topology. 18: 2859–2873 (2018)journal articl

Non-trivalent graph cocycle and cohomology of the long knot space

In this paper we show that via the configuration space integral construction a non-trivalent graph cocycle can also yield a non-zero cohomology class of the space of higher (and even) codimensional long knots. This simultaneously proves that the Browder operation induced by the operad action defined by R. Budney is not trivial.Comment: 17 pages, 11 figures (v2: a comment on a work of R. Longoni is added. v3: Remark 3.6 of v2 has been removed since it might be wrong, as pointed out by I. Volic. We work on R^n instead of the cylinder. Sections 2.2 and 3.3 have been widely revised. Many other minor revisions and corrections.

A New Global Partnering Production Model NGP-PM Utilizing Advanced TPS

In order to improve quality at leading manufacturers’ overseas production bases from the perspective of “global production,” “a New Global Partnering Production Model, NGP-PM” - the strategic development of the “Advanced TPS” model - is proposed and its effectiveness discussed. The aim is to increase global quality by generating a synergetic effect that organically connects and promotes continual evolution of the production plants in Japan and overseas, as well as greater cooperation among production operators