Arithmeticity of groups of fibre homotopy equivalence classes

1980 ◽  
Vol 31 (4) ◽  
pp. 413-424 ◽  
Author(s):  
Hans Scheerer
Keyword(s):  
Equivalence Classes ◽  
Homotopy Equivalence

The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions II

1992 ◽  
Vol 122 (1-2) ◽  
pp. 127-135 ◽  
Author(s):  
John W. Rutter
Keyword(s):  
Group Extension ◽  
Equivalence Classes ◽  
Homotopy Groups ◽  
Geometric Proof ◽  
Homotopy Equivalence ◽  
Connected Space ◽  
Abelian Kernel ◽  
Simply Connected ◽  
Extension Sequence ◽  
Connected Spaces

SynopsisWe give here an abelian kernel (central) group extension sequence for calculating, for a non-simply-connected space X, the group of pointed self-homotopy-equivalence classes . This group extension sequence gives in terms of , where Xn is the nth stage of a Postnikov decomposition, and, in particular, determines up to extension for non-simplyconnected spaces X having at most two non-trivial homotopy groups in dimensions 1 and n. We give a simple geometric proof that the sequence splits in the case where is the generalised Eilenberg–McLane space corresponding to the action ϕ: π1 → aut πn, and give some information about the class of the extension in the general case.

The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions

1992 ◽  
Vol 120 (1-2) ◽  
pp. 47-60 ◽  
Author(s):  
J. W. Rutter
Keyword(s):  
Fundamental Group ◽  
Unit Group ◽  
Group Extension ◽  
Equivalence Classes ◽  
Homotopy Groups ◽  
Homotopy Equivalence ◽  
Special Cases ◽  
Simply Connected ◽  
Extension Sequence ◽  
Connected Spaces

SynopsisWe give here a group extension sequence for calculating, for a non-simply-connected space X, the group of self-homotopy-equivalence classes which induce the identity automorphism of the fundamental group, that is the kernel of the representation → aut (π1(X)). This group extension sequence gives in terms of , where Xn is the n-th stage of a Postnikov decomposition. As special cases, we calculate for non-simply-connected spaces having at most two non-trivial homotopy groups, in dimensions 1 and n, as the unit group of a semigroup structure on ; and we calculate up to extension for non-simply-connected spaces having at most three non-trivial homotopy groups. The group is, for nice spaces, isomorphic to the groups and of self-homotopy-equivalence classes of X in the categories top*M and top M, respectively, where X→M = K1(π1(X)) is a top fibration which determines an isomorphism of the fundamental group; and our results are obtained initially in topM.

scholarly journals Strong and weak (1, 2, 3) homotopies on knot projections

2015 ◽  
Vol 26 (09) ◽  
pp. 1550069 ◽  
Author(s):  
Noboru Ito ◽  
Yusuke Takimura
Keyword(s):  
Weak Type ◽  
Equivalence Classes ◽  
Dimensional Sphere ◽  
Strong Type ◽  
Homotopy Equivalence ◽  
Reidemeister Moves ◽  
Generic Immersion ◽  
Type 3

A knot projection is an image of a generic immersion from a circle into a two-dimensional sphere. We can find homotopies between any two knot projections by local replacements of knot projections of three types, called Reidemeister moves. This paper defines an equivalence relation for knot projections called weak (1, 2, 3) homotopy, which consists of Reidemeister moves of type 1, weak type 2, and weak type 3. This paper defines the first nontrivial invariant under weak (1, 2, 3) homotopy. We use this invariant to show that there exist an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections. By contrast, all equivalence classes of knot projections consisting of the other variants of a triple type, i.e. Reidemeister moves of (1, strong type 2, strong type 3), (1, weak type 2, strong type 3), and (1, strong type 2, weak type 3), are contractible.

scholarly journals Shifted Poisson Structures on Differentiable Stacks

2020 ◽  
Author(s):  
Francesco Bonechi ◽  
Nicola Ciccoli ◽  
Camille Laurent-Gengoux ◽  
Ping Xu
Keyword(s):  
Differential Geometry ◽  
Morita Equivalence ◽  
Equivalence Classes ◽  
Poisson Structures ◽  
Homotopy Equivalence ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Isomorphism Classes ◽  
Relevant Notion ◽  
Differentiable Stack

Abstract The purpose of this paper is to investigate $(+1)$-shifted Poisson structures in the context of differential geometry. The relevant notion is that of $(+1)$-shifted Poisson structures on differentiable stacks. More precisely, we develop the notion of the Morita equivalence of quasi-Poisson groupoids. Thus, isomorphism classes of $(+1)$-shifted Poisson stacks correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following program, which is of independent interest: (1) We introduce a ${\mathbb{Z}}$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under the Morita equivalence of Lie groupoids, and thus they can be considered to be polyvector fields on the corresponding differentiable stack ${\mathfrak{X}}$. It turns out that $(+1)$-shifted Poisson structures on ${\mathfrak{X}}$ correspond exactly to elements of the Maurer–Cartan moduli set of the corresponding dgla. (2) We introduce the notion of the tangent complex $T_{\mathfrak{X}}$ and the cotangent complex $L_{\mathfrak{X}}$ of a differentiable stack ${\mathfrak{X}}$ in terms of any Lie groupoid $\Gamma{\rightrightarrows } M$ representing ${\mathfrak{X}}$. They correspond to a homotopy class of 2-term homotopy $\Gamma$-modules $A[1]\rightarrow TM$ and $T^{\vee } M\rightarrow A^{\vee }[-1]$, respectively. Relying on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids, we prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${\mathfrak{X}}$ defines a morphism ${L_{\mathfrak{X}}}[1]\to{T_{\mathfrak{X}}}$.

Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Main Tool ◽  
Homotopy Equivalence ◽  
Classifying Space ◽  
Deformation Retract ◽  
Simplicial Sets ◽  
Full Proof

Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

scholarly journals THE SYMMETRIC SIGNATURE OF A WITT SPACE

2013 ◽  
Vol 05 (02) ◽  
pp. 121-159 ◽  
Author(s):  
GREG FRIEDMAN ◽  
JAMES McCLURE
Keyword(s):  
Intersection Homology ◽  
Homotopy Equivalence ◽  
New Construction

Witt spaces are pseudomanifolds for which the middle-perversity intersection homology with rational coefficients is self-dual. We give a new construction of the symmetric signature for Witt spaces which is similar in spirit to the construction given by Miščenko for manifolds. Our construction has all of the expected properties, including invariance under stratified homotopy equivalence.

scholarly journals MET: a Java package for fast molecule equivalence testing

2020 ◽  
Vol 12 (1) ◽  
Author(s):  
Jördis-Ann Schüler ◽  
Steffen Rechner ◽  
Matthias Müller-Hannemann
Keyword(s):  
Chemical Properties ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Equivalence Classes ◽  
Equivalence Testing ◽  
Equivalence Test ◽  
Graph Isomorphism Problem ◽  
2D Structure ◽  
Node Labels ◽  
Labelled Graphs

AbstractAn important task in cheminformatics is to test whether two molecules are equivalent with respect to their 2D structure. Mathematically, this amounts to solving the graph isomorphism problem for labelled graphs. In this paper, we present an approach which exploits chemical properties and the local neighbourhood of atoms to define highly distinctive node labels. These characteristic labels are the key for clever partitioning molecules into molecule equivalence classes and an effective equivalence test. Based on extensive computational experiments, we show that our algorithm is significantly faster than existing implementations within , and . We provide our Java implementation as an easy-to-use, open-source package (via GitHub) which is compatible with . It fully supports the distinction of different isotopes and molecules with radicals.

Parallel and Distributed Processability of Objects

1989 ◽  
Vol 12 (3) ◽  
pp. 317-356
Author(s):  
David C. Rine
Keyword(s):  
Software Engineering ◽  
Software Design ◽  
Equivalence Classes ◽  
Minimum Amount ◽  
Software Components ◽  
Limited Extent ◽  
Design Environment ◽  
Distributed Software

Partitioning and allocating of software components are two important parts of software design in distributed software engineering. This paper presents two general algorithms that can, to a limited extent, be used as tools to assist in partitioning software components represented as objects in a distributed software design environment. One algorithm produces a partition (equivalence classes) of the objects, and a second algorithm allows a minimum amount of redundancy. Only binary relationships of actions (use or non-use) are considered in this paper.

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