I had the benefit of Marge Pratt's TeX expertise; In addition to her always patient and thoughtful attention, she typed the book excellently. This is the 'CW complement', namely the largest subcomplex of X whose 0-skeleton consists of the vertices of X that are not in A.
ALGEBRAIC TOPOLOGY FOR GROUP THEORY
CW Complexes and Homotopy
Review of general topology
However, there is a canonical method to "correct" the situation: for any Hausdorf spaceX, one defineskX is the setX endowed with the weak topology with respect to the compact subspaces of X. If X and Y are spaces, we denote by C(X, Y) the set of all maps X →Y endowed with the compact-open topology: for every compact subset K of X and every open subset U of Y let K, U denote the set of all f ∈ C(X, Y) such that f(K) ⊂ U; the family of all such sets K, U is a subbasis for this topology.
CW complexes
A CW complex consists of a space X and a sequence {Xn | n ≥ 0} of subspaces such that. ii) Forn≥1,Xn is obtained from Xn−1 by attaching n cells;. iv) X has the weak topology6 with respect to {Xn}. ThenX×Y (with the product topology understood in the sense ofk-spaces) is a CW complex.
Exercises
Homotopy
If this is further weakened by requiring only that it be homotopic to idAthenAis weak deformation attraction of X. When A is a subcomplex of the CWX complex, the three notions of deformation attraction coincide; see exercises 2 and 3 below.
Maps between CW complexes
Therefore, according to the discussion above, g(Vi) can be covered by the counted number of cubes Cij, so that. Such a path exists because C(d) is a connected path.) Assume that H is already defined on (Xn−1∪A)×I with the desired properties.
Exercise
Neighborhoods and complements
The entire subcomplex of X generated by the vertices X0−A0 is the CW complement of A, denoted by 16X −c A. If Y is a subcomplex and A⊂Y, we write NY(A) for the CW neighborhood of Ain Y, dropping the index when there is no ambiguity.
Cellular Homology
Review of chain complexes
The chain map f :C→D is a chain homotopy equivalence if there is a chain map g:D→C such that g◦f andf◦g are chain homotopy for the respective identity chain maps (of C and of D). This long exact sequence is of course in the sense that if the following diagram of chain cards commute.
Review of singular homology
Singular homology satisfies the excision property that if clXU ⊂ intXA then the inclusion map (X−U, A−U)→(X, A) induces an isomorphism of homology modules. An immediate consequence of this, using homotopy invariance, is that if U ⊂V ⊂AwithU as above and if the pair (X−V, A−V) is a strong deformation attraction of the pair (X −U, A− U), then the excision map (X −V, A−V) → (X, A) also induces an isomorphism of homology modules.
Cellular homology: the abstract theory
The relation φ thus defines a homomorphism ¯φ : Hn∆(Xn+1;R)→Hn(X;R) which can easily be seen as an isomorphism. We have now seen that single homology can be calculated from the cellular chain complex.
The degree of a map from a sphere to itself
Statement 2.4.12. For every open coverU ofIn+1 there exists sk0 such that for everyk≥k0 every cell of Ikn+1 lies in an element of U. Describe a complex CW structure for Rn. for those familiar with category theory) Show that "wedge" is the category-theoretic coproduct of the category Pointed Spaces.
Orientation and incidence number
We can use this to define a preferred orientation on the CW complex as follows: whenn= 0, the canonical orientation is 1; whenn= 1, the canonical orientation onI1 (with two vertices and one 1-cell) is: 1 on each vertex, and (the equivalence class of) id :I1→I1on the 1-cell. Let h : In → In be the restriction of the linear automorphism of Rn which is the identity in all but the last two coordinates and is given by the matrix [0 11 0] in those coordinates. Then this orientation turns on I•n (compare the proof of 2.5.13), but matches the above orientations on the (n−1) planes.
The geometric cellular chain complex
Thus (Cngeom(X;R), ∂) is a chain complex and is canonical chain isomorphic to (Cn(X;R), ∂) as defined in Sect. We have now achieved our goal of defining cellular homology in terms of oriented cells, incidence numbers, and map scales. Prove the following from the definitions given in this section, without using the equivalent formulation in Sect.
Some properties of cellular homology
These are instructive in that they show why it is not easy, perhaps impossible, to give a geometrically independent treatment of cellular homology—a treatment that avoids singular homology altogether. If:X→Y is homotopic to a constant map then f#:Hn(X;R)→Hn(Y;R) is the zero homomorphism when >0. In fact, we can consider H∗( · ;R) as a covariant function from the category of oriented CW complexes and homotopic classes of maps to the category of scaled modules R and degree 0 homomorphisms.
Further properties of cellular homology
The same algebraic trick used to deduce the homology sequence of the oriented CW pair (X,A) can also be used to deduce an exact sequence expressing the homology of a CW complex in terms of the homology of two subcomplexes covering it. This uniqueness theorem, which we will not prove (see for example [61]), implies that any 'homology theory' h∗ satisfying these axioms – for example singular homology – corresponds to cellular homology on oriented finite CW pairs. We have seen in this chapter that the cellular homology H∗( · ;R), with the connecting homomorphism∂∗, satisfies this.
Reduced homology
We reiterate that the reduced homology of nonempty CW complexes differs only from the homology in dimension 0; for the empty CW complex the change occurs in dimension-1. Let X be a CW complex and letY be obtained from X by attaching the ann-cell to Xn−1 using the attachment map :Sn−1 →Xn−1. Set analogous formulas when simultaneous merging (in the sense of Section 1.2) of a group of cells occurs.
Fundamental Group and Tietze Transformations
Combinatorial fundamental group, Tietze transformations, Van Kampen Theorem
Statement 3.1.6. The definition of edge path equivalence is independent of the orientation chosen for X. This homomorphism is indeed a special case of “the homomorphism caused by a map”, but that definition is best left to Section 3.1.6. In a sense, π1(X, v) is independent of the base point v, provided that X has a path.
Appendix: Presentations
Combinatorial description of covering spaces
The quotient mapping q : Y → G\Y =: X is a covering projection and X admits the structure of a complex CW whose cells are {q(e)|e cell Y}. Conversely, given the path of a connected CW complex X, we now show how to construct a simply connected freeπ1(X, v)- CW complex ˜X with quotientX. And as before, the freeπ-action on ˜X1 is extended to make ˜X2 a freeπ-CW complex for which p2 is the quotient map.
Appendix: Cayley graphs
Review of the topologically defined fundamental groupgroup
We will see in the next section that when X is a CW complex, the two definitions of fundamental group agree. There the action seemed to depend on the CW complex structure, on the orientations of the cells and on a maximal tree. Ifp: ˜Y →Y is a covering projection, the group of covering transformationsG (being a group of homeomorphisms) acts to the left on ˜Y; this is independent of the choice of ˜y.
Equivalence of the two definitions of the fundamental group of a CW complexfundamental group of a CW complex
Assertion 3.4.7. (i) Creates a locally connected path; (ii) if Y is obtained from the space connected to the local path A by attaching n-cells, then Y is locally path connected; (iii) each CW complex is locally connected path. So, up to equivalence, there exists a bijection between the vertex-covering spaces of the connected path of the vertex-complex CW (X, v) and the subgroups of π1 (X, v), such that the fundamental group of the covering space corresponding to H is isomorphically mapped to H. Describe a CW complex X that has a vertex v, a 1-cell, and a 2-cell such that X− {v} is a path-connected space, the fundamental set of which has not been definitively created.
Some Techniques in Homotopy Theory
Altering a CW complex within its homotopy type
Theorem 4.1.10 Let P be obtained from P by a Tietze transformation of Type I. Then there is a homotopy equivalence, which leads the following diagram to homotopy: There are homotopy equivalent CW complexes YP1 and YP2. obtained from XP1 and XP2 by attaching 3 cells. Gis presented finitely, there is a CW complex X, obtained from iii) If G is presented finitely,
Appendix: the equivariant case
- Cell trading
- Domination, mapping tori, and mapping telescopes
- Review of homotopy groups
- Geometric proof of the Hurewicz Theorem
Give a counterexample to "Whitehead's Theorem" in Exercise 1 when X does not have a homotopic type of the CW complex. Thus, X is an example of a finite CW complex whose second homotopic group is not finitely generated (as an abelian group). Find an example of a path-connected CW complex X where h1 is an isomorphism but2 is not surjective.
Elementary Geometric Topology
- Review of topological manifolds
- Simplicial complexes and combinatorial manifolds
- Regular CW complexes
- Incidence numbers in simplicial complexes
If M bears the structure of a CW complex, then the dimension (in the sense of Section 1.2) of that CW complex does not. Using this, prove that every CW complex X is the homotopy type of some |J| has where J is an abstract simple complex. We have just seen that in a regular CW complex an n-cell has planes of all lower dimensions.
FINITENESS PROPERTIES OF GROUPSGROUPS
The Borel Construction and Bass-Serre Theory
The Borel construction, stacks, and rebuilding
If for each cell e of C we are given a CW complex Fe of the same homotopy type as Fe, then there is a stack of CW complexes π. The content of Theorem 6.1.3 is that q:Z→V is a stack of CW complexes with base spaceV, in which the fiber over the cell line is Ge˜\X. We can use 6.1.4˜ to replace G˜e\X˜ by a more desirable CW complex of the same homotopy type and thereby produce a more desirable space Z of the same homotopy type as Z . The first of a number of uses of this method appears in:. Note that (1-cell ofXe)×(vertex ofB1) does not give a new 1-cell of the addition complex.) So Z has a finite 1-skeleton.
Decomposing groups which act on trees (Bass-Serre Theory)Theory)
Associated with (G, Γ) and T is a group, denoted π1(G, Γ;T), called the fundamental group of (G, Γ) based on T, namely: the quotient of the free product. This group is the HNN extension of Gbyφ; the subgroup G (see 6.2.1) is the base group and is called the stable letter. Example 6.2.9. The group SL2(Z) of 2×2 integer matrices of determinant 1 acts on the open upper half of the complex plane Cby M¨obius transformations.
Appendix: Generalized graphs of groups
The vertex and edge stabilizers of this new graph of groups (¯G, Γ) are the images in the vertex and edge groups of G. Describe the Bass-Serre tree in the circle of groups corresponding to the given presentation of the Baumslag-Solitar group BS (m, n). Give an example with G finitely presented such that ker(α) is not generated finitely; give another graph of groups decomposition of the same group G with a vertex and an edge such that the vertex and edge groups are finitely generated.
Topological Finiteness Properties and Dimension of Groups
- K ( G, 1) complexes
- Finiteness properties and dimensions of groups
- Recognizing the finiteness properties and dimension of a groupof a group
- Brown’s Criterion for finiteness
Claim 7.1.3. A path-connected CW complex X is not aspherical if its universal cover X˜ is n-connected. X is aspheric if X˜ is contractible. 6 This is where two uses of the letter F can cause confusion: "type F" and "group F." Claim 7.2.17. If there exists an aK(G,1)-complex dominated by a d-dimensional CW complex, then G×Z has geometric dimension≤d+ 1.
Homological Finiteness Properties of Groups
- Homology of groups
- Homological finiteness properties
- Synthetic Morse theory and the Bestvina-Brady Theorem
- Affine CW complexes
- Morse Theory
- Finiteness properties and Morse Theory
- Prescribing finiteness properties
Thus ˜f# is a chain homotopy equivalence of RG chain complexes.3 The chain homotopy equivalence constructed in the last proof induces f˜∗:H0( ˜X;R)→H0(˜Y;R). We list some well-known properties of a convex cell, which have been proved in many books (e.g. [90, Ch. 1]): (i) the faces of C are determined by C and not by the choice of fi'er and gj' is used to define C ; (ii) C is the convex hull of its vertices; (iii) the image of C under an affine map RN →RM is a convex cell; (iv) there is a simple complexK in RN such that C = |K|; (v) writing C• for the union of all proper faces of the compact convex cell C, there is a PL homeomorphism (C,C)• →(∆n,∆• n) - i.e. C is a PLn sphere. A proper CAT(0) space is a metric space (M, d) with the following properties: (i) it is a non-empty geodesic metric space: this means that an isometric copy of the closed interval [0, d( a, b)] called ageodesic segment connects two arbitrary points a, b ∈ M; (ii) for any geodesic triangle ∆ in M with vertices a, b, c let ∆ denote a triangle in the Euclidean plane with vertices a, b, c and corresponding side lengths of ∆ and ∆ equal; let ω and ω isometrically parametrize geodesic segments from btoc and frombtoc, respectively; then for any 0≤t≤d(b, c),d(a, ω(t))≤ |a−ω(t)|; and (iii) incorrect, i.e. the closed sphere around any a∈M of any radius rice compact.
