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Lecture Notes on Homotopy Theory and Applications

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Lecture Notes on Homotopy Theory and Applications LAURENTIUMAXIM UNIVERSITYOFWISCONSIN-MADISON LECTURENOTES ONHOMOTOPYTHEORY ANDAPPLICATIONS i Contents 1 Basics of Homotopy Theory 1 1.1 Homotopy Groups 1 1.2 Relative Homotopy Groups 7 1.3 Homotopy Extension Property 10 1.4 Cellular Approximation 11 1.5 Excision for homotopy groups. The Suspension Theorem 13 1.6 Homotopy Groups of Spheres 13 1.7 Whitehead’s Theorem 16 1.8 CW approximation 20 1.9 Eilenberg-MacLane spaces 25 1.10 Hurewicz Theorem 28 1.11 Fibrations. Fiber bundles 29 1.12 More examples of fiber bundles 34 1.13 Turning maps into fibration 38 1.14 Exercises 39 2 Spectral Sequences. Applications 41 2.1 Homological spectral sequences. Definitions 41 2.2 Immediate Applications: Hurewicz Theorem Redux 44 2.3 Leray-Serre Spectral Sequence 46 2.4 Hurewicz Theorem, continued 50 2.5 Gysin and Wang sequences 52 ii 2.6 Suspension Theorem for Homotopy Groups of Spheres 54 2.7 Cohomology Spectral Sequences 57 2.8 Elementary computations 59 n 2.9 Computation of pn+1(S ) 63 3 2.10 Whitehead tower approximation and p5(S ) 66 Whitehead tower 66 3 3 Calculation of p4(S ) and p5(S ) 67 2.11 Serre’s theorem on finiteness of homotopy groups of spheres 70 2.12 Computing cohomology rings via spectral sequences 74 2.13 Exercises 76 3 Fiber bundles. Classifying spaces. Applications 79 3.1 Fiber bundles 79 3.2 Principal Bundles 86 3.3 Classification of principal G-bundles 92 3.4 Exercises 97 4 Vector Bundles. Characteristic classes. Cobordism. Applications. 99 4.1 Chern classes of complex vector bundles 99 4.2 Stiefel-Whitney classes of real vector bundles 102 4.3 Stiefel-Whitney classes of manifolds and applications 103 The embedding problem 103 Boundary Problem. 107 4.4 Pontrjagin classes 109 Applications to the embedding problem 112 4.5 Oriented cobordism and Pontrjagin numbers 113 4.6 Signature as an oriented cobordism invariant 116 4.7 Exotic 7-spheres 117 4.8 Exercises 118 iii List of Figures Figure 1.1: f + g ......................... 2 Figure 1.2: f + g ' g + f .................... 2 Figure 1.3: f + g, revisited................... 3 Figure 1.4: bg .......................... 3 Figure 1.5: relative bg ...................... 9 Figure 1.6: universal cover of S1 _ Sn ............. 15 Figure 1.7: The mapping cylinder M f of f .......... 19 Figure 2.1: r-th page Er ..................... 42 Figure 2.2: n-th diagonal of E¥ ................ 42 Figure 2.3: p-axis and q-axis of E2 .............. 44 basics of homotopy theory 1 1 Basics of Homotopy Theory 1.1 Homotopy Groups Definition 1.1.1. For each n ≥ 0 and X a topological space with x0 2 X, the n-th homotopy group of X is defined as n n pn(X, x0) = f : (I , ¶I ) ! (X, x0) / ∼ where I = [0, 1] and ∼ is the usual homotopy of maps. Remark 1.1.2. Note that we have the following diagram of sets: f (In, ¶In) / (X, x ) 7 0 g ( (In/¶In, ¶In/¶In) n n n n n with (I /¶I , ¶I /¶I ) ' (S , s0). So we can also define n pn(X, x0) = g : (S , s0) ! (X, x0) / ∼ . Remark 1.1.3. If n = 0, then p0(X) is the set of connected components 0 0 of X. Indeed, we have I = pt and ¶I = Æ, so p0(X) consists of homotopy classes of maps from a point into the space X. Now we will prove several results analogous to the case n = 1, which corresponds to the fundamental group. Proposition 1.1.4. If n ≥ 1, then pn(X, x0) is a group with respect to the operation + defined as: 8 1 < f (2s1, s2,..., sn) 0 ≤ s1 ≤ 2 ( f + g)(s , s ,..., sn) = 1 2 1 :g(2s1 − 1, s2,..., sn) 2 ≤ s1 ≤ 1. (Note that if n = 1, this is the usual concatenation of paths/loops.) Proof. First note that since only the first coordinate is involved in this operation, the same argument used to prove that p1 is a group is valid 2 homotopy theory and applications In−1 Figure 1.1: f + g f g 0 1/2 1 s1 here as well. Then the identity element is the constant map taking all n of I to x0 and the inverse element is given by − f (s1, s2,..., sn) = f (1 − s1, s2,..., sn). Proposition 1.1.5. If n ≥ 2, then pn(X, x0) is abelian. Intuitively, since the + operation only involves the first coordinate, if n ≥ 2, there is enough space to “slide f past g”. Figure 1.2: f + g ' g + f f f g ' f g ' ' g f g ' g f Proof. Let n ≥ 2 and let f , g 2 pn(X, x0). We wish to show that f + g ' g + f . We first shrink the domains of f and g to smaller cubes n inside I and map the remaining region to the base point x0. Note that this is possible since both f and g map to x0 on the boundaries, so the resulting map is continuous. Then there is enough room to slide f past g inside In. We then enlarge the domains of f and g back to their original size and get g + f . So we have “constructed” a homotopy between f + g and g + f , and hence pn(X, x0) is abelian. Remark 1.1.6. If we view pn(X, x0) as homotopy classes of maps n (S , s0) ! (X, x0), then we have the following visual representation of f + g (one can see this by collapsing boundaries in the above cube interpretation). basics of homotopy theory 3 Figure 1.3: f + g, revisited f c X g Next recall that if X is path-connected and x0, x1 2 X, then there is an isomorphism bg : p1(X, x1) ! p1(X, x0) where g is a path from x1 to x0, i.e., g : [0, 1] ! X with g(0) = x1 and g(1) = x0. The isomorphism bg is given by bg([ f ]) = [g¯ ∗ f ∗ g] −1 for any [ f ] 2 p1(X, x1), where g¯ = g and ∗ denotes path concatana- tion. We next show a similar fact holds for all n ≥ 1. Proposition 1.1.7. If n ≥ 1 and X is path-connected, then there is an isomorphism bg : pn(X, x1) ! pn(X, x0) given by bg([ f ]) = [g · f ], where g is a path in X from x1 to x0, and g · f is constructed by first shrinking the domain of f to a smaller cube inside In, and then inserting the path g radially from x1 to x0 on the boundaries of these cubes. x0 Figure 1.4: bg x1 x0 x1 f x1 x0 x1 x0 Proof. It is easy to check that the following properties hold: 1. g · ( f + g) ' g · f + g · g 2. (g · h) · f ' g · (h · f ), for h a path from x0 to x1 3. cx0 · f ' f , where cx0 denotes the constant path based at x0. 4. bg is well-defined with respect to homotopies of g or f . Note that (1) implies that bg is a group homomorphism, while (2) and (3) show that bg is invertible. Indeed, if g(t) = g(1 − t), then −1 bg = bg. 4 homotopy theory and applications So, as in the case n = 1, if the space X is path-connected, then pn is independent of the choice of base point. Further, if x0 = x1, then (2) and (3) also imply that p1(X, x0) acts on pn(X, x0) as: p1 × pn ! pn (g, [ f ]) 7! [g · f ] Definition 1.1.8. We say X is an abelian space if p1 acts trivially on pn for all n ≥ 1. In particular, this implies that p1 is abelian, since the action of p1 on p1 is by inner-automorphisms, which must all be trivial. We next show that pn is a functor. Proposition 1.1.9. A map f : X ! Y induces group homomorphisms f∗ : pn(X, x0) ! pn(Y, f(x0)) given by [ f ] 7! [f ◦ f ], for all n ≥ 1. Proof. First note that, if f ' g, then f ◦ f ' f ◦ g. Indeed, if yt is a homotopy between f and g, then f ◦ yt is a homotopy between f ◦ f and f ◦ g. So f∗ is well-defined. Moreover, from the definition of the group operation on pn, it is clear that we have f ◦ ( f + g) = (f ◦ f ) + (f ◦ g). So f∗([ f + g]) = f∗([ f ]) + f∗([g]). Hence f∗ is a group homomorphism. The following is a consequence of the definition of the above induced homomorphisms: Proposition 1.1.10. The homomorphisms induced by f : X ! Y on higher homotopy groups satisfy the following two properties: 1. (f ◦ y)∗ = f∗ ◦ y∗. 2. (id ) = id . X ∗ pn(X,x0) We thus have the following important consequence: Corollary 1.1.11. If f : (X, x0) ! (Y, y0) is a homotopy equivalence, then f∗ : pn(X, x0) ! pn(Y, f(x0)) is an isomorphism, for all n ≥ 1. Example 1.1.12. Consider Rn (or any contractible space). We have n n pi(R ) = 0 for all i ≥ 1, since R is homotopy equivalent to a point. The following result is very useful for computations: Proposition 1.1.13. If p : Xe ! X is a covering map, then p∗ : pn(Xe, xe) ! pn(X, p(xe)) is an isomorphism for all n ≥ 2. Proof. First we show that p∗ is surjective. Let x = p(xe) and consider n n f : (S , s0) ! (X, x).
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