Connected cw complex pdf

This conjecture is interesting and highly nontrivial even in the case where the group g acts trivially on the space m. Suppose that x is a connected cw complex such that for some n41 there is a map a as above. Thus for each cell in a, the image of its attaching map is contained in a, so a is itself a cw complex. After all cells have been attached one has a cw complex z with a weak homotopy equivalence f.

Its cw complex topology is the same as the topology induced from x, as one sees by noting inductively that. In other words, what is special about the homotopy type of a compact smooth manifold m. The precise definition prescribes how the cells may be topologically glued together. Here we collect a number of basic topological facts about cw complexes for con venient reference. Andrew blumberg, equivariant homotopy theory, 2017 pdf, github a standard reference is. If x is a space, then a subspace of x is a subset a. By a theorem in class, a cw complex is connected if and only if it is path connected. The doldthom is not, however, merely of interest as a technical curio connecting homology and homotopy theory. Show that every cw complex is locally path connected. Theres a key theorem about cw complexes, that the inclusion of any of any subcomplex into the entire cwcomplex is a cofibration. We can put on sn the cellular structure consisting of a single 0cell, say e0, together with a single ncell, say en. It is a cw complex made of nondegenerate simplices of the simplicial set. For a simply connected cwcomplex x of dimension n, let y be a space obtained by attaching cells of dimension q to x, q n, and let ay denote an adamshilton model of y. Thus every connected cw complex is homotopy equivalent to a cw complex with these additional properties.

Remark origin of the cw terminology the terminology cwcomplex goes back to john henry constantine whitehead and see the discussion in hatcher, topology of cell complexes, p. X characteristic maps, for in an arbitrary index set and n 2. Conclude by showing that the colimit of a sequence of closed inclusions of locally path connected spaces is locally path connected. For a cw complex with finitely many cells, the euler characteristic can be defined as. 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.

Note that the cwcomplex constructed in this proof is also pathconnected, so the correspondence theorem applies and so there is a pathconnected nsheeted cover p. The rational toral rank r0x of a space x is the largest integer rsuch that the torus tr can act continuously on a cwcomplex in the rational homotopy type of x with all its isotropy subgroups. A nite complex is a cwcomplex with only nitely many cells. Simply connected and 4dimensional cw complex, homotopy selfequivalences, whitehead exact sequence, sequences, automorphisms 1. How do you prove a cw complex is locally path connected. First show that attaching cells to a locally path connected space produces a locally path connected space. For example, suppose x is a pathconnected cwcomplex with only even dimensional cells. Observe that if x is a nite complex, a4 is redundant, since w is the union of the compact sets.

The simplices can be glued to themselves or multiply glued to each other. This is given by sending a g gcw complex, y y, to the presheaf sending g h gh to y h yh, the subspace of y y fixed by h h see at elmendorfs theorem references. Since it is a cwcomplex, x is locally pathconnected and semilocally simplyconnected because it is locally contractible. A space is connected if it cannot be written as the disjoint union. The homotopy hypothesis generalizing to weak ngroupoids. Xsuch that the total space eis homotopy equivalent to a nite dimensional cwcomplex. Theres a key theorem about cw complexes, that the inclusion of any of any subcomplex into the entire cw complex is a cofibration. Connectedpdf is a document conversion application that converts standard pdfs into shareable pdfs that can be updated even after being sent out. The collection of g gcwcomplexes has a full embedding into the infinity,1presheaves on the orbit category orb g orbg. The cohomology of xis periodic if and only if there is an orientable spherical bration e. Assume that xhas the homotopy type of a connected cwcomplex. Simply connected and 4dimensional cwcomplex, homotopy selfequivalences, whitehead exact sequence, sequences, automorphisms 1.

Thus, if a subset of a cwcomplex is closed, then a mapping from the topological space into a topological space is continuous if and only if the restrictions of to the. The euler characteristic of a space with finitely generated homology is the lefschetz number also called lefschetz trace of the identity map from to itself. X, where y 1 is a point in the interior of the cell di. For a simply connected cw complex x of dimension n, let y be a space obtained by attaching cells of dimension q to x, q n, and let ay denote an adamshilton model of y. If x is pathconnected, this procedure produces a cw approximation having a single 0 cell. Look at that proof and the neighbourhoods constructed in that proof. For an example of a connected, uncountable cw complex, you can for example take this one.

Cw complexes soren hansen this note is meant to give a short introduction to cw complexes. Equilibrium points in nperson games social sciences. Suppose x,x0 is a connected cw complex note that cw complexes are lo cally path connected, so that this implies x is path connected. The c stands for closurefinite, and the w for weak topology an ndimensional closed cell is the image of an ndimensional closed ball under an attaching map. A nite complex is a cw complex with only nitely many cells. A gcw complex y, b is a subcomplex of x, a if y is a gsubspace of x, bis a closed gsubspace of a, and y n y\x n in the cw decomposition. Since a cwcomplex is a colimit in top over its cells, and as such equipped with the final topology of the cell inclusion maps, a subset of a cwcomplex is open or closed precisely if its restriction to the closure of each cell is open or closed, respectively. There is a spectral sequence whose initial term is, in low dimensions, the homology groupl of f with coefficients in the homotopy group1 of x and whose final term is the graded group of the homotopy group of x y with respect to a suitable filtration. Whitehead torsion, part ii lecture 4 harvard university. The new version is automatically recorded in the cloud in an intuitive document version tree. Then m g is a closed, smooth, simplyconnected, spinable 5manifold with h 2m g. However, the simplicial set structure also implies that the corners of each simplex are consistently locally ordered, and this is not possible with an arbitrary gluing. A b e a relativ e g cw complex and let y b e aspace suc h that h is nonempt y, connected, and nsimple if h o ccurs as an isotrop y subgroup of x a. First show that attaching cells to a locally pathconnected space produces a locally pathconnected space.

When does there exist a homotopy equivalence xm, where m is a compact smooth manifold. X which is a union of cells of x, such that the closure of each cell in a is contained in a. Conclude by showing that the colimit of a sequence of closed inclusions of locally pathconnected spaces is locally pathconnected. If eay denotes the group of homotopy selfequivalences of ay and. It is obviously natural with respect to maps of the pair y, b, when attention is paid to basepoints, and it commutes with injections and boundary. Then the boundary maps in the cellular chain complex are. Solutions to homework 2 attaching one cell, we could pick a basepoint x 0 in the interior of the cell and take a y. Its cw complex topology is the same as the topology induced from x, as one sees by noting inductively that the two topologies agree on an a. Introduction lecture 1 february 2, 2011 in this course, we will be concerned with variations on the following. The class of cwcomplexes or the class of spaces of the same homotopy type as a cwcomplex is the most suitable class of topological spaces in relation to homotopy theory. The term cw complex comes from \closure nite with the weak topology, where \closure nite refers to a3 and \weak topology refers to a4. X y between topological spaces x and y is a function which is continuous.

The simplest cw complex, complex, regular complex, and simplicial complex homeomorphic to the torus. Show that every cwcomplex is locally pathconnected. Contents gcw complexes d g x a i university of chicago. In topology, a cw complex is a type of topological space introduced by j. Cohomology groups of lens spaces uwmadison department of. Then for any connected cwcomplex x, we have theorem 1 the doldthom theorem. Cinduces isomorphisms on all homotopy groups, but it is not a homotopy equivalence, so the cw hypothesis is required. Y with y pathconnected, show that fcan be extended to a map x. Show that as a result, if x has the homotopy type of a nite cwcomplex, xis contractible.

Let c be the cantor set with the discrete topology. A discrete group acts freely and properly on rn sm for some. The term cwcomplex comes from \closure nite with the weak topology, where \closure nite refers to a3 and \weak topology refers to a4. Pdf it is shown that the hyperspace of a connected cwcomplex is an absolute retract for stratifiable spaces, where the hyperspace is the space of. A further feature which can be useful is that all the attaching maps for the cells of z are basepointpreserving. Our result has a number of consequences, which we now list. We have decided to use the term cellular space instead of cw complex, following. Show that a cw complex retracts onto any contractible subcomplex. Denote x the regular covering space corresponding to n. A g cw complex y, b is a subcomplex of x, a if y is a gsubspace of x, bis a closed gsubspace of a, and y n y\x n in the cw decomposition. Introduction let xbe a cw complex, and let ex denote the group of homotopy classes of selfhomotopy equivalences of x.

Pdf let d be the class of all cw complexes homotopy dominated by finite cw complexes. This is given by sending a g gcw complex, y y, to the presheaf sending g h gh to y h yh, the subspace of y y fixed by h h. Postnikov systems exist on connected cw complexes and there is a weak homotopyequivalence between and cw complex z and a. Notation and conventions in the following a space is a topological space and a map f. The determination of the group ex presents a challenging problem of computation.

Here are some facts about cwcomplexes that we will not prove but will use if need. Postnikov systems exist on connected cw complexes and there is a weak homotopyequivalence between and lecture 1 february 2, 2011 in this course, we will be concerned with variations on the following. Let xbe an aspherical space, and y a connected, nite cwcomplex. Then every element 2whg can be realized as the whitehead torsion of a homotopy equivalence f. The document owner can make changes to the pdf, check a box, and instantly notify the recipients that there is an updated version. Roughly speaking, a cw complex is made of basic building blocks called cells. Show that a cw complex is contractible if it is the union of an. Onthetopologicalsocialchoicemodel university of chicago. In other words, the wedge sum of an uncountable number of circles. One can also apply this technique to produce a cw approximation to a pair x,x0. S2 is pathconnected, and the homology groups of the spheres, we obtain.