scholarly journals The loop space homotopy type of simply-connected four-manifolds and their generalizations

2014 ◽  
Vol 262 ◽  
pp. 213-238 ◽  
Author(s):  
Piotr Beben ◽  
Stephen Theriault
Keyword(s):  
Homotopy Type ◽  
Loop Space ◽  
Four Manifolds ◽  
Simply Connected

scholarly journals On the K-theory of the loop space of a Lie group

1974 ◽  
Vol 76 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Francis Clarke
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Simple Structure ◽  
Homology Theory ◽  
Compact Lie Group ◽  
Classifying Space ◽  
K Theory ◽  
Simply Connected ◽  
Torsion Free ◽  
Multiplicative Structures

Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).

Suspension of the Lusternik-Schnirelmann Category

1970 ◽  
Vol 22 (6) ◽  
pp. 1129-1132
Author(s):  
William J. Gilbert
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Category Structure ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Cw Complexes ◽  
Lusternik Schnirelmann ◽  
Image Position ◽  
Simply Connected ◽  
Homotopy Invariants

Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],We prove thatandIt is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.

RACK SPACES AND LOOP SPACES

1999 ◽  
Vol 08 (01) ◽  
pp. 99-114 ◽  
Author(s):  
Bert Wiest
Keyword(s):  
Homotopy Type ◽  
Loop Space ◽  
Topological Spaces ◽  
Loop Spaces

We prove that the rack and quandle spaces of links in 3-manifolds, considered only as topological spaces (disregarding their cubical structure), are closely related to certain subspaces of the loop spaces on the 3-manifold, which we call the vertical and the straight loop space of the link. Using these models we prove that the homotopy type of the non-augmented rack and quandle spaces of a link L in a 3-manifold M depends essentially only on the homotopy type of the pair (M,M -L).

scholarly journals Le probl\`eme de la sch\'ematisation de Grothendieck revisit\'e

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Bertrand Toën
Keyword(s):  
Homotopy Type ◽  
Homotopy Groups ◽  
Simply Connected

The objective of this work is to reconsider the schematization problem of [6], with a particular focus on the global case over Z. For this, we prove the conjecture [Conj. 2.3.6][15] which gives a formula for the homotopy groups of the schematization of a simply connected homotopy type. We deduce from this several results on the behaviour of the schematization functor, which we propose as a solution to the schematization problem. Comment: 21 pages, french

scholarly journals Einstein Four-Manifolds With Self-Dual Weyl Curvature of Nonnegative Determinant

2019 ◽  
Author(s):  
Peng Wu
Keyword(s):  
Scalar Curvature ◽  
Positive Scalar Curvature ◽  
The Self ◽  
Weyl Curvature ◽  
Positive Scalar ◽  
Four Manifolds ◽  
Simply Connected

Abstract We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally Kähler if and only if the determinant of the self-dual Weyl curvature is positive.

ON POSITIVELY CURVED FOUR-MANIFOLDS WITH S1-SYMMETRY

2011 ◽  
Vol 22 (07) ◽  
pp. 981-990 ◽  
Author(s):  
JIN HONG KIM
Keyword(s):  
Topological Classification ◽  
Circle Action ◽  
Torus Action ◽  
Dimensional Manifold ◽  
Linear Action ◽  
Circle Actions ◽  
Four Manifolds ◽  
Simply Connected ◽  
Diffeomorphism Classification ◽  
Positively Curved

It is well known by the work of Hsiang and Kleiner that every closed oriented positively curved four-dimensional manifold with an effective isometric S1-action is homeomorphic to S4 or CP2. As stated, it is a topological classification. The primary goal of this paper is to show that it is indeed a diffeomorphism classification for such four-dimensional manifolds. The proof of this diffeomorphism classification also shows an even stronger statement that every positively curved simply connected four-manifold with an isometric circle action admits another smooth circle action which extends to a two-dimensional torus action and is equivariantly diffeomorphic to a linear action on S4 or CP2. The main strategy is to analyze all possible topological configurations of effective circle actions on simply connected four-manifolds by using the so-called replacement trick of Pao.

On an Algebra Structure on the Hochschild Homology of Simply Connected Topological Space

2019 ◽  
Vol 26 (03) ◽  
pp. 425-436
Author(s):  
Calvin Tcheka
Keyword(s):  
Topological Space ◽  
Commutative Algebra ◽  
Loop Space ◽  
Hochschild Homology ◽  
Algebra Structure ◽  
Free Loop Space ◽  
Graded Algebras ◽  
Simply Connected ◽  
Space Cohomology ◽  
Improved Outcome

In this note, we use the pairing induced by the interchange map in conjunction with the strongly homotopy commutative algebra structure to define products on the Eilenberg–Moore differential Tor and give a simplified proof of an improved outcome of Jones’s result due to Ndombol and Thomas. As a result, we establish an isomorphism of graded algebras between the Hochschild homology and the free loop space cohomology of a simply connected topological space.

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