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Exercises 1 CW-Complexes, Homotopy Extension Property

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Exercises 1 CW-Complexes, Homotopy Extension Property Exercises 1 CW-complexes, homotopy extension property 1. (Rated G) Hatcher no. 23 of chapter 0 on contractible complexes. 2. (G) Suppose X is a CW-complex, A a subcomplex. Then X=A is a CW-complex with cells corresponding to the cells of X that are not in A, plus an additional 0-cell corresponding to A itself (collapsed to a point). 3. (PG-13) This long exercise would be best done by a team of people. It is an illustration of the use of direct limit topologies, plus some interesting applications. Let X be a CW-complex, given as the union of an ascending chain of subcomplexes X0 ⊂ X1 ⊂ :::. In fact the CW-structure per se is not relevant; the key point for the exercise is that X has the direct limit topology with respect to the Xi's, and X × I has the direct limit topology with respect to the Xi × I's (the latter statement following from Hatcher Theorem A6, since I is compact). I make it a CW-complex to avoid getting distracted by the general and very technical question of how direct limit and product topologies interact. a) Suppose that Xn−1 is a retract of Xn for all n. Show that X0 is a retract of X by making sense of “infinite composition" of the given retractions. b) Now suppose Xn−1 is a deformation retract of Xn for all n. Show that X0 is a deformation retract of X by making sense of “infinite stacking" of homotopies in this setting. Model your argument on the one used by Hatcher in Prop. 1A.1 on maximal trees in graphs. c) Understand the proof of the aforementioned Prop 1A1 (the trickiest step being already done in part (b) above) and explain it to the rest of us. d) Show that any CW-complex X is homotopy-equivalent to a CW-complex Y having only one 0-cell. In particular, if X is 1-dimensional then Y is a wedge of circles. e) Maybe look at some further applications, such as the topological proof that every subgroup of a free group is free (see Hatcher 1A.4) and tell us about them. 4. (PG) This problem, a brief introduction to Schubert cells, comes in real and complex flavors; to be definite let's stick to the complex case. Consider the natural linear action of n n−1 GLnC on C . It induces an evident transitive action on CP , which can be regarded as a topological group action on a space, or better as a smooth Lie group action on a smooth manifold. (For fans of algebraic geometry, it is even an algebraic group action on a projective variety.) If H ⊂ GLnC is any subgroup, we get an H-action by restriction. The relevant subgroup for this problem in Bn := BnC, the group of upper triangular invertible matrices. n−1 a) Show that the cells of the standard CW-structure (as in Hatcher) on CP are pre- cisely the Bn-orbits. (For fans of algebraic geometry: the cells are then locally Zariski-closed 1 subvarieties, isomorphic to affine spaces of the appropriate dimension.) Identify the closures of the cells, showing in particular that the boundary of each cell is a union of cells of lower dimension (recall that the definition of CW-complex only requires that the boundary is con- tained in a union of cells of lower dimension). Note the explicit description of the attaching maps given in Hatcher. n b) More generally consider the Grassmannian GkC . Interpret the standard column reduction/column echelon form (columns are more appropriate here than rows) for n × k matrices, which many of you will teach at some point in Math 308, as showing that the Bn- n orbits of the natural action on GkC are homeomorphic to complex vector spaces and hence to even-dimensional cells. These are the Schubert cells. (The above remark for algebraic geometers applies here too.) c) The cell decomposition in (b) is indeed a CW-decomposition, but more work is needed to exhibit characteristic maps. As an optional project, look at Milnor's \Characteristic Classes" to see how this is done. It's \elementary", but rather involved. The fact that the cells are all even-dimensional will make it easy to \compute" the homology groups later. d) Check that it all works in the real case as well (although the cells are no longer all even-dimensional). 5. (PG) This problem assumes familiarity with tubular neighborhoods. Suppose M is a smooth manifold and N ⊂ M is a closed embedded submanifold. Then (M; N) has the homotopy extension property. In fact one can give (M; N) the structure of a CW-pair, or even a simplicial pair (to be defined later), but the proof is very complicated and technical. Your mission is to give a simpler and more direct proof of the HEP based on \mapping cylinder neighborhoods" (Hatcher Example 0.15). n 6. (PG-13) Show that every open subset of R can be given the structure of a CW- complex. 2 Homology I 1. x2.2 of Hatcher exercises 1-8 on the degree of a map Sn −! Sn are all good, but especially take a look at 2,3,8. Problem 7 is discussed further in (4) below. 2. Exercise 20 of Hatcher x2.1 is in fact one of the most important theorems of the subject (the suspension isomorphism). 3. Show that there is a long exact sequence for the reduced homology of a pair (X; A), agreeing with the one on ordinary homology in dimensions ≥ 1. Thus the only thing you ~ have to check is that @ : H1(X; A) −! H0A in fact has image in H0A, and that ~ ~ H1(X; A) −! H0A −! H0X −! H0(X; A) −! 0 2 is exact. This is elementary and might seem overly fussy to bother with, but it's worth paying attention to reduced homology. It simplifies both the statements and the proofs of various results, as we'll see. 4. Here are two important theorems that may or may not be familiar: Theorem 2.1 a) The orthogonal group O(n) has two path-components, distinguished by the determinant ±1. b) The general linear group GLnR has two path-components, distinguished by the sign of the determinant. + c) Multiplication O(n) × Bn R −! GLnR is a homeomorphism, and hence O(n) is a + deformation retract of GLnR. Here Bn R denotes the upper triangular matrices with positive diagonal entries, and is contractible. Note that thanks to part (c), parts (a) and (b) are equivalent. The second theorem is just the complex analogue. Theorem 2.2 a) The unitary group U(n) is path-connected. b) The general linear group GLnC is path-connected. + c) Multiplication U(n) × Bn C −! GLnC is a homeomorphism, and hence U(n) is a + deformation retract of GLnR. Here Bn C denotes the upper triangular matrices with positive real diagonal entries, and is contractible. It's highly recommended to prove these results yourself. There are several different ways to do it, but all of them boil down to linear algebra. First of all properties (c) are little more than reformulations of the Gram-Schmidt orthogonalization process. Then you have your choice of proving properties (a) or properties (b). The idea is to either use some canonical form or another for matrices (there are several choices that work), or some type of factorization (e.g. orthogonal matrices can be factored as a product of reflections) to find a path from your matrix to the identity matrix or (in the real case) to a reflection matrix. Using these theorems one can easily compute the degree of a linear map. By this we mean either (a) a map Sn −! Sn given by an orthogonal or unitary matrix, or (b) a general linear map in the situation of Hatcher x2.2 Exercise 7. The former case will be done in class; the latter is left as the exercise. 3 Homology II n 1. The complex conjugation map on CP is defined by σ([z0; :::; zn]) = [z0; :::zn]. Check that this is a well-defined continuous (indeed smooth) map, and show that it is not homotopic to the identity map. You would of course do this by showing the induced map on homology is not the identity. While you're at it, even though this is more than you need to answer the n original question, compute the induced map explicitly on every H2kCP . n m 2. A real truncated projective space is a space of the form RP =RP , where m < n. 3 Note: These spaces are more important than you might think. In class I'll outline how they come up in the vector field problem for Sn. The vector field application involves the concept of \reducibility" defined in part (b) below. a) Use the cellular chain complex to compute the homology of all such spaces (making n use of the case m = 0 that we've already done, i.e. RP itself, to make things easier). n m n n−1 ∼ n b) Let π : RP =RP −! RP =RP = S denote the evident quotient map. We say n m n n m that RP =RP is reducible if there is a map f : S −! RP =RP such that π ◦ f is a homotopy equivalence. An informal, more evocative terminology is to say that \the top cell splits off". n m Show that if RP =RP is reducible then n is odd. (The converse is false, but we don't yet have the means to show this.) 3.
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