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1.
Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely
maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As
a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences
between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S
n+1
, n ⩾ 0.
The first author was partially supported by NSERC research grant.
Received December 12, 2001; in revised form September 7, 2002
Published online February 28, 2003 相似文献
2.
We show that the n-homotopy category of connected (n+1)-dimensional Menger manifolds is isomorphic to the homotopy category of connected Hilbert cube manifolds whose k-dimensional homotopy groups are trivial for each . 相似文献
3.
We show that the n-homotopy category of connected (n+1)-dimensional Menger manifolds is isomorphic to the homotopy category of connected Hilbert cube manifolds whose k-dimensional homotopy groups are trivial for each .
(Received 30 August 1999; in revised form 7 December 1999) 相似文献
4.
The closed model category of exterior spaces, that contains the proper category, is a useful tool for the study of non compact
spaces and manifolds. The notion of exterior weak ℕ-S-equivalences is given by exterior maps which induce isomorphisms on the k-th ℕ-exterior homotopy groups for k ∈ S, where S is a set of non negative integers. The category of exterior spaces with a base ray localized by exterior weak ℕ-S-equivalences is called the category of exterior ℕ-S-types. The existence of closed model structures in the category of exterior spaces permits to establish equivalences between
homotopy categories obtained by dividing by exterior homotopy relations, and categories of fractions (localized categories)
given by the inversion of classes of week equivalences. The family of neighbourhoods ‘at infinity’ of an exterior space can
be interpreted as a global prospace and under the condition of first countable at infinity we can consider a global tower
instead of a prospace. The objective of this paper is to use localized categories to find the connection between S-types of exterior spaces and S-types of global towers of spaces. The main result of this paper establishes an equivalence between the category of S-types of rayed first countable exterior spaces and the category of S-types of global towers of pointed spaces. As a consequence of this result, categories of global towers of algebraic models
localized up to weak equivalences can be used to give some algebraic models of S-types.
The authors acknowledge the financial support given by the projects FOMENTA 2007/03 and MTM2007-65431. 相似文献
5.
Pierre Ghienne 《manuscripta mathematica》2002,107(3):289-310
Zabrodsky exact sequences are algebraic tools which express the genus set of a space X in term of its self-maps, when X has the rational homotopy type of a co-ℋ-space or an ℋ-space. Explicit examples show these methods can't be generalized to
the class of all simply connected finite CW-complexes.
We however construct a Zabrodsky exact sequence for those three cells CW-complexes rationally equivalent to the product of
two spheres S
k
×S
n
, n>k≥2. We deduce, from results of Morisugi-Oshima, the genus of some spherical bundles.
Received: 17 March 2001 / Revised version: 8 August 2001 相似文献
6.
Hans-Joachim Baues 《K-Theory》1996,10(2):107-133
Let n+k>n2. We consider algebraic models of the homotopy category of CW-complexes with cells only in dimension n and n+k. The algebraic model category is described in terms of non-Abelian extensions in Ext2. 相似文献
7.
We give a necessary condition for Morita equivalence of simple Generalized Weyl algebras of classical type. We propose a reformulation
of Hodges’ result, which describes Morita equivalences in case the polynomial defining the Generalized Weyl algebra has degree
2, in terms of isomorphisms of quantum tori, inspired by similar considerations in noncommutative differential geometry. We
study how far this link can be generalized for n ≥ 3. 相似文献
8.
A classC of pointed spaces is called a cellular class if it is closed under weak equivalences, arbitrary wedges and pointed homotopy
pushouts. The smallest cellular class containingX is denoted byC(X), and a partial order relation ≪ is defined by:X ≪Y ifY εC(X). In this text we investigate the sub partial order sets generated respectively by simply connected finite CW-complexes and
by rational spaces. For rational spaces we prove a unique decomposition theorem, a density theorem and the existence of infinitely
many non-comparable elements. We then prove the density theorem for a generic class of finite CW-complexes. 相似文献
9.
Let k be an algebraically closed field. Let Λ be the path algebra over k of the linearly oriented quiver
\mathbb An\mathbb A_n for n ≥ 3. For r ≥ 2 and n > r we consider the finite dimensional k −algebra Λ(n,r) which is defined as the quotient algebra of Λ by the two sided ideal generated by all paths of length r. We will determine for which pairs (n,r) the algebra Λ(n,r) is piecewise hereditary, so the bounded derived category D
b
(Λ(n,r)) is equivalent to the bounded derived category of a hereditary abelian category H\mathcal H as triangulated category. 相似文献
10.
We study additive representability of orders on multisets (of size k drawn from a set of size n) which satisfy the condition of independence of equal submultisets (IES) introduced by Sertel and Slinko (Ranking committees,
words or multisets. Nota di Laboro 50.2002. Center of Operation Research and Economics. The Fundazione Eni Enrico Mattei,
Milan, 2002, Econ. Theory 30(2):265–287, 2007). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete
cones. Following Fishburn (J. Math. Psychol., 40:64–77, 1996) and Conder and Slinko (J. Math. Psychol., 48(6):425–431, 2004), we define functions f(n,k) and g(n,k) which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable
order satisfying the IES, respectively, from a representable order. We prove that g(n,k) = n − 1 whenever n ≥ 3 and (n, k) ≠ (5, 2). In the exceptional case, g(5,2) = 3. We also prove that g(n,k) ≤ f(n,k) ≤ n and establish that for small n and k the functions g(n,k) and f(n,k) coincide.
相似文献
11.
M. Šilhavý 《Milan Journal of Mathematics》2008,76(1):275-306
The paper gives a decomposition of a general normal r-dimensional current [5] into the sum of three measures of which the first is an r-dimensional rectifiable measure, the second is the Cantor part of the current, and the third is Lebesgue absolutely continuous.
This is analogous to the well-known decomposition of the derivative of a function of bounded variation into the jump, Cantor,
and absolutely continuous parts; in fact the last is a special case of the result for (n–1)-dimensional normal currents. Further, Whitney’s cap product [15] is recast in the language of the approach to flat chains
by Federer [5] and a special case (viz., currents of dimension n – 1) is shown to be closely related to the measure-valued duality pairings between vector measures with curl a measure and
L∞ vectorfields with L∞ divergence as established by Anzellotti [2] and Kohn & Témam [6]. Finally, the cap product is shown to be jointly weak* continuous
in the two factors of the product in a way similar to the compensated compactness theory; in the cases of (n – 1)-dimensional objects this reduces to results closely related to the div–curl lemmas of the standard compensated compactness
theory.
Received: June 2007 相似文献
12.
Thomas Hüttemann 《Geometriae Dedicata》2010,148(1):175-203
Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U
σ
= Spec A
σ
. A quasi-coherent sheaf on X gives rise, by taking sections over the U
σ
, to a diagram of modules over the coordinate rings A
σ
, indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent
sheaves on X can be obtained from a category of Σop-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits.
Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the
machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves
in the sense that chain complexes over different open sets U
σ
agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves
is equivalent to the derived category of X. 相似文献
13.
Norbert H. Schlomiuk 《Annali di Matematica Pura ed Applicata》1972,92(1):211-215
Summary In[4] we have defined a notion of μ-homotopy in the category of simplicial groups and we have made the conjecture that μ-homotopy
is equivalent to loop homotopy. The purpose of this paper is to prove this cojecture.
Entrata in Redazione il 25 febbraio 1972.
This work was done while the author held a Visiting Professorship at the Istituto Matematico, Università di Perugia, under
the auspices of the Italian National Research Council. 相似文献
14.
In this paper, the authors give the L
p
(1 < p < ∞ ) boundedness of the k-th order commutator of parabolic singular integral with the kernel function Ω ∈ L(log +
L)
k + 1(S
n − 1). The result in this paper is an extension of some known results.
The research was supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025). 相似文献
15.
For a family F{{\cal F}} of subsets of [n] = {1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that F{{\cal F}} is P-free if it does not contain a subposet isomorphic to P. Let ex(n, P) be the largest size of a P-free family of subsets of [n]. Let Q
2 be the poset with distinct elements a, b, c, d, a < b,c < d; i.e., the 2-dimensional Boolean lattice. We show that 2N − o(N) ≤ ex(n, Q
2) ≤ 2.283261N + o(N), where
N = \binomn?n/2 ?N = \binom{n}{\lfloor n/2 \rfloor}. We also prove that the largest Q
2-free family of subsets of [n] having at most three different sizes has at most 2.20711N members. 相似文献
16.
Luis Javier Hernández Paricio 《Applied Categorical Structures》2005,13(5-6):421-451
For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences
are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees
higher than or equal to n. We take coefficients either on all the cyclic groups with s ∈ S, or in the abelian group where is the group of fractions of the form with s ∈ S. In the first case, for n > 1 the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion. In the second case, for n > 1 we obtain that the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion and the nth homotopy group is divisible. These equivalences of categories are given by colocalizations , obtained by cofibrant approximations on the model structures. These colocalization maps have nice universal properties. For
instance, the map is final (in the homotopy category) among all the maps of the form Y → X with Y an (n − 1)-connected CW-complex whose homotopy groups are S-torsion and its nth homotopy group is divisible. The spaces , are constructed using the cones of Moore spaces of the form M(T, k), where T is a coefficient group of the corresponding structure of models, and homotopy colimits indexed by a suitable ordinal. If
S is generated by a set P of primes and S
p
is generated by a prime p ∈ P one has that for n > 1 the category is equivalent to the product category . If the multiplicative system S is generated by a finite set of primes, then localized category is equivalent to the homotopy category of n-connected Ext-S-complete CW-complexes and a similar result is obtained for . 相似文献
17.
W. Malfait 《Monatshefte für Mathematik》2001,133(2):157-162
We show that from dimension six onwards (but not in lower dimensions), there are in each dimension flat manifolds with first
Betti number equal to zero admitting Anosov diffeomorphisms. On the other hand, it is known that no flat manifolds with first
Betti number equal to one support Anosov diffeomorphisms. For each integer k > 1 however, we prove that there is an n-dimensional flat manifold M with first Betti number equal to k carrying an Anosov diffeomorphism if and only if M is a k-torus or n is greater than or equal to k + 2.
(Received 5 October 2000; in revised form 9 March 2001) 相似文献
18.
Simone Borghesi 《Inventiones Mathematicae》2003,151(2):381-413
For any prime q and positive integer t, we construct a spectrum k(t) in the stable homotopy category of schemes over a field k equipped with an embedding k↪ℂ. In classical homotopy theory, the ℂ realization of k(t) is known as Morava K-theory. The algebraic content lies in the fact that these spectra are defined as the homotopy limit
of a tower whose cofibers are appropriate suspensions of the motivic Eilenberg-MacLane spectra, which are known to represent
motivic cohomology in the stable homotopy category of schemes.
Oblatum 26-XI-2001 & 5-VIII-2002?Published online: 8 November 2002 相似文献
19.
Let k be a field and E(n) be the 2
n+1-dimensional pointed Hopf algebra over k constructed by Beattie, Dăscălescu and Grünenfelder [J. Algebra, 2000, 225: 743–770]. E(n) is a triangular Hopf algebra with a family of triangular structures R
M
parameterized by symmetric matrices M in M
n
(k). In this paper, we study the Azumaya algebras in the braided monoidal category $
E_{(n)} \mathcal{M}^{R_M }
$
E_{(n)} \mathcal{M}^{R_M }
and obtain the structure theorems for Azumaya algebras in the category $
E_{(n)} \mathcal{M}^{R_M }
$
E_{(n)} \mathcal{M}^{R_M }
, where M is any symmetric n×n matrix over k. 相似文献
20.
We consider a category \({\mathcal H}^{\ominus \otimes}\) (the homotopy category of homotopy squares) whose objects are homotopy commutative squares of spaces and whose morphisms are cubical diagrams subject to a coherent homotopy relation. The main result characterises the isomorphisms of \({\mathcal H}^{\ominus \otimes}\) to be the cube morphisms whose forward arrows are homotopy equivalences. As a first application of the new category we give a direct 2-track theoretic definition of the quaternary Toda bracket operation. 相似文献