scholarly journals A zoo of growth functions of mapping class sets

2018 ◽  
Vol 12 (03) ◽  
pp. 841-855 ◽  
Author(s):  
Fedor Manin
Keyword(s):  
Mapping Class ◽  
Rational Homotopy ◽  
Growth Functions ◽  
Lipschitz Maps ◽  
Rational Cohomology ◽  
Rational Homotopy Type ◽  
Number Formula ◽  
Mapping Classes ◽  
Simply Connected ◽  
The Universe

Suppose [Formula: see text] and [Formula: see text] are finite complexes, with [Formula: see text] simply connected. Gromov conjectured that the number of mapping classes in [Formula: see text] which can be realized by [Formula: see text]-Lipschitz maps grows asymptotically as [Formula: see text], where [Formula: see text] is an integer determined by the rational homotopy type of [Formula: see text] and the rational cohomology of [Formula: see text]. This conjecture was disproved in a recent paper of the author and Weinberger; we gave an example where the “predicted” growth is [Formula: see text] but the true growth is [Formula: see text]. Here we show, via a different mechanism, that the universe of possible such growth functions is quite large. In particular, for every rational number [Formula: see text], there is a pair [Formula: see text] for which the growth of [Formula: see text] is essentially [Formula: see text].

scholarly journals CONTINUOUS TRACE C*-ALGEBRAS, GAUGE GROUPS AND RATIONALIZATION

2009 ◽  
Vol 01 (03) ◽  
pp. 261-288 ◽  
Author(s):  
JOHN R. KLEIN ◽  
CLAUDE L. SCHOCHET ◽  
SAMUEL B. SMITH
Keyword(s):  
Gauge Group ◽  
Metric Space ◽  
Principal Bundle ◽  
Rational Homotopy ◽  
Compact Metric Space ◽  
Gauge Groups ◽  
C Algebra ◽  
C Algebras ◽  
Rational Cohomology ◽  
Rational Homotopy Type

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let Aζdenote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UAζ, the group of unitaries of Aζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

scholarly journals The extended rational homotopy theory of operads

2018 ◽  
Vol 25 (4) ◽  
pp. 493-512
Author(s):  
Benoit Fresse
Keyword(s):  
Homotopy Theory ◽  
American Mathematical Society ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Graded Algebras ◽  
Simplicial Sets ◽  
Differential Graded Algebras ◽  
Rational Cohomology ◽  
Rational Homotopy Type ◽  
Set Up

Abstract In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book [B. Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups. Part 2. The Applications of (Rational) Homotopy Theory Methods, Math. Surveys Monogr. 217, American Mathematical Society, Providence, 2017]. In short, we prove that the rational homotopy type of such an operad is determined by a cooperad in cochain differential graded algebras (a cochain Hopf dg-cooperad for short) as soon as the Sullivan rational homotopy theory works for the spaces underlying our operad (e.g. when these spaces are connected, nilpotent, and have finite-type rational cohomology groups).

scholarly journals Rational Models of the Complement of a Subpolyhedron in a Manifold with Boundary

2018 ◽  
Vol 70 (2) ◽  
pp. 265-293 ◽  
Author(s):  
Hector Cordova Bulens ◽  
Pascal Lambrechts ◽  
Don Stanley
Keyword(s):  
Configuration Space ◽  
Compact Manifold ◽  
Algebraic Model ◽  
Configuration Spaces ◽  
Rational Homotopy ◽  
Homotopy Invariance ◽  
Rational Models ◽  
Manifold With Boundary ◽  
Rational Homotopy Type ◽  
Simply Connected

AbstractLet W be a compact simply connected triangulated manifold with boundary and let K ⊂ W be a subpolyhedron. We construct an algebraic model of the rational homotopy type of W\K out of a model of the map of pairs (K, K⋂∂W) ↪ (W, ∂W) under some high codimension hypothesis.We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicit models of these configuration spaces for a large class of compact manifolds.

CARDINALITY OF RATIONAL HOMOTOPY TYPES OF SIMPLY CONNECTED CW COMPLEXES

2011 ◽  
Vol 22 (02) ◽  
pp. 179-193
Author(s):  
MAHMOUD BENKHALIFA
Keyword(s):  
Exact Sequence ◽  
Finite Type ◽  
Equivalence Class ◽  
Graded Algebra ◽  
Rational Homotopy ◽  
Homotopy Types ◽  
Cw Complex ◽  
Rational Homotopy Type ◽  
Cw Complexes ◽  
Simply Connected

The aim of this paper is to consider the following problem: given a simply connected CW complex X of finite type, find the cardinal of the set of rational homotopy types of CW complexes Y such that H*(Y, ℚ) is isomorphic, as a commutative graded algebra, to H*(X, ℚ). For this purpose we introduce a long exact sequence, called the Whitehead exact sequence, which we associated with the minimal Sullivan model of X and which allows us to define the notion of the adapted couple associated with X. Thus we show that the rational homotopy type of X is completely determined by the equivalence class of its adapted couple.

l′-Isolated Maps and Localizations

1983 ◽  
Vol 35 (2) ◽  
pp. 193-217
Author(s):  
Sara Hurvitz
Keyword(s):  
Finite Number ◽  
Free Algebra ◽  
Nilpotent Groups ◽  
Homotopy Groups ◽  
Homology Groups ◽  
Rational Cohomology ◽  
Locally Nilpotent Groups ◽  
Definition Of ◽  
Simply Connected ◽  
Path Connected

Let P be the set of primes, l ⊆ P a subset and l′ = P – l Recall that an H0-space is a space the rational cohomology of which is a free algebra.Cassidy and Hilton defined and investigated l′-isolated homomorphisms between locally nilpotent groups. Zabrodsky [8] showed that if X and Y are simply connected H0-spaces either with a finite number of homotopy groups or with a finite number of homology groups, then every rational equivalence f : X → Y can be decomposed into an l-equivalence and an l′-equivalence.In this paper we define and investigate l′-isolated maps between pointed spaces, which are of the homotopy type of path-connected nilpotent CW-complexes. Our definition of an l′-isolated map is analogous to the definition of an l′-isolated homomorphism. As every homomorphism can be decomposed into an l-isomorphism and an l′-isolated homomorphism, every map can be decomposed into an l-equivalence and an l′-isolated map.

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