287 research outputs found
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: 2859â2873 (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 manufacturersâ overseas production bases from the perspective of âglobal production,â âa New Global Partnering Production Model, NGP-PMâ - the strategic development of the âAdvanced TPSâ 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
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