共查询到20条相似文献，搜索用时 15 毫秒
1.
Alexander Dranishnikov Jerzy Dydak 《Transactions of the American Mathematical Society》2001,353(1):133-156
The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results:
Generalized Eilenberg-Borsuk Theorem. Let be a countable CW complex. If is a separable metrizable space and is an absolute extensor of for some CW complex , then for any map , closed in , there is an extension of over an open set such that .
Theorem. Let be countable CW complexes. If is a separable metrizable space and is an absolute extensor of , then there is a subset of such that and .
Theorem. Suppose are countable, non-trivial, abelian groups and 0$\">. For any separable metrizable space of finite dimension 0$\">, there is a closed subset of with for .
Theorem. Suppose is a separable metrizable space of finite dimension and is a compactum of finite dimension. Then, for any , , there is a closed subset of such that and .
Theorem. Suppose is a metrizable space of finite dimension and is a compactum of finite dimension. If and are connected CW complexes, then
2.
The notion of (strongly) hereditarily aspherical compacta introduced by Daverman (1991) is modified. The main results are: Theorem. If is a hereditarily aspherical compactum, then ANR. In particular, is strongly hereditarily aspherical. Theorem. Suppose is a cell-like map of compacta and is shape aspherical for each closed subset of . Then
- 1.
- Y is hereditarily shape aspherical,
- 2.
- is a hereditary shape equivalence,
- 3.
- .
- 1.
- and ,
- 2.
- .
3.
Let be a countable and locally finite CW complex. Suppose that the class of all metrizable compacta of extension dimension contains a universal element which is an absolute extensor in dimension . Our main result shows that is quasi-finite.
4.
Alex Chigogidze 《Proceedings of the American Mathematical Society》2000,128(7):2187-2190
We show that for each countable simplicial complex the following conditions are equivalent:
- iff for any space .
- There exists a -invertible map of a metrizable compactum with onto the Hilbert cube.
5.
Leonard R. Rubin 《Proceedings of the American Mathematical Society》1997,125(10):3125-3128
Approximate (inverse) systems of compacta have been useful in the study of covering dimension, dim, and cohomological dimension over an abelian group , . Such systems are more general than (classical) inverse systems. They have limits and structurally have similar properties. In particular, the limit of an approximate system of compacta satisfies the important property of being an approximate resolution. We shall prove herein that if is an abelian group, a compactum is the limit of an approximate system of compacta , , and for each , then .
6.
Jerzy Dydak 《Topology and its Applications》2004,140(2-3):227-243
We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper:
Theorem. Suppose X is a paracompact space. There is a CW complex K such that
- (a) K is an absolute extensor of X up to homotopy,
- (b) If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy.
Theorem. Let X be a paracompact space. Suppose a space Y is the union of a family {Ys}sS of its subspaces with the following properties:
- (a) Each Ys is an absolute extensor of X,
- (b) For any two elements s and t of S there is uS such that YsYtYu.
If f :A→Y is a map from a closed subset A to Y such that A=sSIntA(f−1(Ys)), then f extends over X.
That result implies a few well-known theorems of classical theory of retracts which makes it of interest in its own. 相似文献
7.
A. Dranishnikov 《Geometriae Dedicata》2009,141(1):59-86
We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension
asdim
Z
X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology
is strictly less than its asymptotic dimension.
相似文献
8.
Akira Koyama Manuel A. Moron 《Proceedings of the American Mathematical Society》2002,130(10):3091-3096
We shall prove the following: Let be a refinable map between paracompact spaces. Then is finitistic if and only if is finitistic. Let be a hereditary shape equivalence between metric spaces. Then if is finitistic, is finitistic.
9.
We introduce the notion of a (stable) dimension scale d-sc(X) of a space X, where d is a dimension invariant. A bicompactum X is called dimensionally unified if dim F = dimG F for every closed F ? X and for an arbitrary abelian group G. We prove that there exist dimensionally unified bicompacta with every given stable scale dim-sc. 相似文献
10.
Hirotaka Tamanoi 《Transactions of the American Mathematical Society》1997,349(3):1209-1237
Fundamental classes in cohomology of Eilenberg-MacLane spaces are defined. The image of the Thom map from cohomology to mod- cohomology is determined for arbitrary Eilenberg-MacLane spaces. This image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of Milnor primitives to the fundamental class in mod- cohomology. This subalgebra in mod cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that cohomology determines Morava cohomology for Eilenberg-MacLane spaces.
11.
Ilijas Farah 《Proceedings of the American Mathematical Society》2002,130(4):1243-1246
We prove that the Cech-Stone remainder of the integers, , maps onto its square if and only if there is a nontrivial map between two of its different powers, finite or infinite. We also prove that every compact space that maps onto its own square maps onto its own countable infinite product.
12.
Grant F. Armstrong Grant Cairns Gunky Kim 《Proceedings of the American Mathematical Society》1999,127(3):709-714
We show that if is a finite dimensional real Lie algebra, then has cohomological dimension if and only if is a unimodular extension of the two-dimensional non-Abelian Lie algebra .
13.
Jerzy Dydak 《Transactions of the American Mathematical Society》2006,358(4):1537-1561
The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory, this algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes of graded groups . There are two geometric interpretations of these equivalence classes: 1) For pointed CW complexes and , if and only if the infinite symmetric products and are of the same extension type (i.e., iff for all compact ). 2) For pointed compact spaces and , if and only if and are of the same dimension type (i.e., for all Abelian groups ).
Dranishnikov's version of the Hurewicz Theorem in extension theory becomes for all simply connected .
The concept of cohomological dimension of a pointed compact space with respect to a graded group is introduced. It turns out iff for all . If and are two positive graded groups, then if and only if for all compact .
14.
Alexander N. Dranishnikov 《Indagationes Mathematicae》2018,29(1):429-449
We give a survey of old and new results in dimension theory of compact metric spaces. Most of the relatively new results presented in the survey are based on the cohomological dimension approach. We complement the survey by stating the basics of cohomological dimension theory and listing some of its applications beyond the dimension theory. 相似文献
15.
16.
Jesús Ferrer 《Journal of Mathematical Analysis and Applications》2002,265(2):322-331
In an infinite-dimensional real Hilbert space, we introduce a class of fourth-degree polynomials which do not satisfy Rolle's Theorem in the unit ball. Extending what happens in the finite-dimensional case, we show that every fourth-degree polynomial defined by a compact operator satisfies Rolle's Theorem. 相似文献
17.
Kamal Bahmanpour 《代数通讯》2019,47(3):1327-1347
In this article, we shall investigate the concepts of cofiniteness of local cohomology modules and Abelian categories of cofinite modules over arbitrary Noetherian rings. Then we shall improve some of the well known results in the area. 相似文献
18.
In [2] Camillo and Zelmanowitz stated that rings all whose modules are dimension modules are semisimple Artinian. It seem however that the proof in [2] contains a gap and applies to rings with finite Goldie dimension only. In this paper we show that the result indeed holds for all rings with a basis as well as for all commutative rings with Goldie dimension attained. 相似文献
19.
Alexander Schmidt 《Compositio Mathematica》2002,133(3):267-288
We remove the assumption p 2 or k is totally imaginary from several well-known theorems on Galois groups with restricted ramification of number fields. For example, we show that the Galois group of the maximal extension of a number field k which is unramified outside 2 has finite cohomological 2-dimension (also if k has real places). 相似文献
20.
Robert Kaufman 《Proceedings of the American Mathematical Society》2000,128(2):427-431
We investigate dimension-increasing properties of maps in Sobolev spaces; we obtain sharp results with a random process somewhat like Brownian motion.