A finite cw complex which admits an a, form admits an a, form not necessarily extending the given a, or a, forms. X inherits a cellular decomposition which makes it a cwcomplex. Every cw complex has the homotopy type of a simplicial complex. Beware however that the terminology clashes somewhat with the use in homotopy theory, there the concept of finite homotopy type is crucially different from homotopy type with finite. This is true in the simply connected case, but seems like a difficult problem in general. The homotopy type of a finite cw complex is called a finite homotopy type. Many authors impose further conditions, often because they are only interested in specific types of rings and not the most general case. Special cases include the classical groups, the chevalley groups, the steinberg groups, and the suzukiree groups. On the homotopy of finite cwcomplexes with polycyclic fundamental group mihai damian universit. Let x be a connected cwcomplex, with only a finite number of ceils in each dimension, and let x, denote its nskeleton.
Free actions of finite groups on finite cw complexes zdziskaw wojtkowiak 1 commentarii mathematici helvetici volume 55, pages 225 232 1980 cite this article. The cw complex is formed by taking a quotient of a compact space the finite union of compacts is compact. The notation cw comes from the initial letters of the english names for the above two conditions c for closure finiteness and w for weak topology. A cw complex is a type of topological space introduced by j.
Now in i a homotopy invariant ring kx was introduced for a finite cw complex x. The real line admits the structure of 1dimensional cw complex with the integers as zerocells and the intervals n. Then the classifying space of the topological monoid of the self homotopy equivalences of y is of finite type i. A gcwcomplex x is of finite type if each n skeleton is finite. The most obvious example is provided by the cohomology ring.
X will always have the homotopy type of a connected. On the homotopy of finite cw complexes with polycyclic fundamental group mihai damian universit. Model of wood cells with 70% porosity in fully aligned case 0% offset of cellular structure with no free water in lumen. We use these results to define a homotopy invariant ia of cwcomplexes for any finite.
The map h is continuous when restricted to any finite skeleton, so is continuous. Amir h assadi the problem that we are concerned with is the existence and construction of embeddings of a given gcw complex gmanifold in another gcw complex g. In geometric topology, a field within mathematics, the obstruction to a finitely dominated space x being homotopyequivalent to a finite cw complex is its wall finiteness obstruction wx which is an element in the reduced zeroth algebraic ktheory of the integral group ring. Finite group actions on simplyconnected manifolds and cw complexes. This addresses the modified question in jeremys comments, on keeping the preferred cwstructure. In fact it has the somewhat stronger property that each of its homotopy groups is of finite type i. In topology, a cw complex is a type of topological space introduced by j. When both conditions are satisfied, there is an element of the projective class group of the integral group ring of. A finite cell complex satisfies both conditions c and w. Cw complexes which are not homotopically stratified by orbit type. Finite z2zcw complexes which are not homotopically stratified by orbit type andrew nicas and david rosenthal abstract. Mod p stable orders of finite cwcomplexes sciencedirect. On the homotopy of closed manifolds and finite cwcomplexes yang su and xiaolei wu abstract. In fact it has the somewhat stronger property that each of.
More generally, a cell complex each point of which is contained in some finite subcomplex is a. A thesis submitted in partial fulfilment of the requirements for the degree of master of science department of 1athematics, statistics and computer science memorial university of newfoundland december, 1977 st. You know via hatcher that theyre dominated by a finite cw complex, hence you can apply walls obstruction theory to being homotopy equivalent to a finite cw complex. X r for some n and an open neighborhood u of ix such that ix.
Els evier topology and its applications 73 1996 233240 topology and its applications mod p stable orders of finite cw complexes huajian yang department of mathematics lehigh university bethlehem, pa 18015, usa received 26 june 1995. Volume 25, number 1, july 1991 elliptic spaces yves felix, stephen halperin, and jeanclaude thomas abstract. Cw complexes which are not homotopically stratified by orbit type article pdf available in proceedings of the american mathematical society 71 january 2009 with 31 reads. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation often with a much smaller complex. A finite cwcomplex is one which admits a presentation in which there are only finitely many attaching maps, and similarly a countable cwcomplex is one which admits a presentation with countably many attaching maps. Stong minimal models of closed surfaces in this section, we construct regular cw models for all closed surfaces and show that the associated posets have no beat points, making them stong minimal. Smooth extensions for finite cw complexes 345 in 2, we generalize definition 0. The term classifying space stems from the fact that isomorphism classes of principal gbundles over a cwcomplex bcorrespond bijectively to homotopy classes of maps b bgsee 34. The invariant was originally formulated in the context of surgery on cw complexes, generalizing swans application of algebraic ktheory to the study of free actions of finite groups on spheres. Sullivan conj ecture 10 asserts that, given a finitedimensional the of pointed maps connected cw complex x and a finite group g, the space x from the classifying space bg to x has the weak homotopy type of a point. A finite cw complex, that is, one with only finitely many cells, is compact since. In, a finite cw complex x is constructed so that x admits an a3 form but no a 5 form. R for some n and an open neighborhood u of ix such that ix.
Using a theorem of bruce hughes, we show that these complexes are not homotopically strati ed by orbit type in the sense of quinn. A group g is called of type fp if the trivial zgmodule zhas a projective resolution of. Topological spaces of the same homotopy type as cwcomplexes are considered cwspaces. Finite cwcomplexes memorial university research repository. Pdf on the homotopy of finite cwcomplexes with polycyclic. A pair x,e consisting of a hausdorff space x and a cell. Accordingly, a chain complex of vector spaces, possibly those generating a semifree dga is said to be of finite type if it is degreewise finite dimensional. Finite z2z cw complexes which are not homotopically stratified by orbit type andrew nicas and david rosenthal abstract. A nite complex is a cwcomplex with only nitely many cells. In geometric topology, a field within mathematics, the obstruction to a finitely dominated space x being homotopyequivalent to a finite cwcomplex is its wall finiteness obstruction wx which is an element in the reduced zeroth algebraic ktheory of the integral group ring. Finite element analysis is a method of solving, usually approximately, certain problems in engineering and science. See 3 for another construction leading to this statement.
In 93, a finite c w complex x is constructed so that x admits an a3 form but no a, form. Topological spaces of the same homotopy type as cw complexes are considered cw spaces. This immediately implies that every simply connected compact manifold is homotopy equivalent to a. On the homotopy type and the fundamental crossed complex of. Construction methods are used to give algebraic characterizations of cw complexes, firstly of finite type, and secondly, of finite dimension. When both conditions are satisfied, there is an element of the projective class group of the integral group ring of the fundamental group of the space, a homotopy type invariant, whose vanishing is necessary and sufficient to guarantee that a cw complex be finite. More generally, a cell complex each point of which is contained in some finite subcomplex is a cwcomplex. An account of the former will be given in paper ii of this series and of the latter in s. Then phx, y 0 for all plocal spaces y iand only if i. The homotopy type of b o is uniquely determined by g, so that homotopy invariants of b o give rise to algebraic invariants 1 of g. The wall finiteness obstruction lecture 2 harvard math. Zp is finite, then by sduality and kunneth theorem, h,x a dx.
Wall by work of john milnor on finitely dominated spaces. A finite cwcomplex is a cwcomplex which admits a presentation in which there are a finite number of attaching maps. Homotopy associativity and finite cw complexes sciencedirect. A difference between us and browns approach is that we avoid the use of a variant of browns lemma, a homologicalalgebra lemma which underlies many results on finitess conditions.
Especially, the results in 17, 11 and 12 are direct consequences of our theorem. A finite celldecomposition is a cell decomposition consisting of finitely many cells. The homotopy type of bgdoes not depend upon the choice of eg. The homotopy of finite cwcomplexes 1793 if x is a manifold with nonzero euler characteristic, the same holds for. We construct a nite cw complex which is shown in another paper 10 to be a k1 for this picture group. As such, it is a numerical rather than an analytical method. Taking a quotient preserves compactness since the quotient space is the image of the original space under the projection map, and continuous maps preserve compactness. Spaces xand y are weakly homotopy equivalent if and only if there is a space zin fact a cw complex z and weak homotopy equivalences z. Picture groups of finite type and cohomology in type a n kiyoshi igusa, kent orr, gordana todorov, and jerzy weyman abstract. This conj ecture was resolved in the affirmative in 9. It is used mainly for problems for which no exact solution, expressible in some mathematical form, is available.
A pointed connected space x9 x of finite shape dimension is pointed stable if and only if each of its homotopy progroups pro7inx9 x is stable, i. Finite groups of lie type give the bulk of nonabelian finite simple groups. A finite cw complex is one which admits a presentation in which there are only finitely many attaching maps, and similarly a countable cw complex is one which admits a presentation with countably many attaching maps. The term cwcomplex comes from \closure nite with the weak topology, where \closure nite refers to a3 and \weak topology refers to a4. In this paper, we shall often use complex for cw complex, and equivalent for homotopy equivalent. Finite cwcomplexes by harold edward alexander campbell, b. Construction methods are used to give algebraic characterizations of cwcomplexes, firstly of finite type, and secondly, of finite dimension. Furthermore, edwards and geoghegan have obtained the following algebraic stability criterion. It is then natural and for infinite dimensional spaces. If x is elliptic it has the homotopy type of a finite poincare complex. This addresses the modified question in jeremys comments, on keeping the preferred cw structure. A group of lie type is a group closely related to the group gk of rational points of a reductive linear algebraic group g with values in the field k. The purpose of the present study is to construct a counterexample to the above conjecture.
For any finite simplicial complex x, there exists an embedding i. A finite cw complex is a cw complex which admits a presentation in which there are a finite number of attaching maps. Gu and hunt2d finite element heat transfer model, moisture effects 161 fig. Let x be a c w complex with finite type and let p be a fixed prime. Does m have the homotopy type of a finite cw complex. Homotopy associativity and finite cw complexes 127 roughly, this means that odd spheres are homotopy associative hspaces mod primes 3. In this section we set up the foundations of proper homoty theory. Of course, if m is triangulable, then it is actually homeomorphic to a finite simplicial.
Observe that if x is a nite complex, a4 is redundant, since w is the union of the compact sets. A simply connected cw complex of finite type is elliptic if it has finite l. This and theorem 6 below lead to an algebraic description of the 3type of any complex and of the 4type of any simplyconnected, finite complex. Self homotopy equivalences of virtually nilpotent spaces. The homotopy type of a finite cwcomplex is called a finite homotopy type. When both conditions are satisfied, there is an element of the projective class group of the integral group ring of the fundamental group of the space, a homotopy. Methods of this type are needed because analytical.
Free actions of finite groups on finite cw complexes. Walls finiteness obstruction is an algebraic ktheory invariant which decides if a finitely dominated space is homotopy equivalent to a finite cw complex. A finite cw complex which admits an a3 form admits an a form not necessarily extending the given az or a3 forms. Lectures on the cohomology of finite groups 3 2 using joins, we may construct a model for egwhich is functorial in g, namely eg colim i g.