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Algebraic Topology Andrew Kobin Spring { Fall 2016 Contents Contents Contents 0 Introduction 1 0.1 Differential Forms . .1 0.2 The Exterior Derivative . .6 0.3 de Rham Cohomology . .8 0.4 Integration of Differential Forms . 12 1 Homotopy Theory 18 1.1 Homotopy . 18 1.2 Covering Spaces . 28 1.3 Classifying Covering Spaces . 36 1.4 The Fundamental Theorem of Covering Spaces . 43 1.5 The Seifert-van Kampen Theorem . 45 2 Homology 51 2.1 Singular Homology . 51 2.2 Some Homological Algebra . 61 2.3 The Eilenberg-Steenrod Axioms . 66 2.4 CW-Complexes . 75 2.5 Euler Characteristic . 85 2.6 More Singular Homology . 87 2.7 The Mayer-Vietoris Sequence . 94 2.8 Jordan-Brouwer Separation Theorem . 98 2.9 Borsuk-Ulam Theorem . 100 2.10 Simplicial Homology . 103 2.11 Lefschetz's Fixed Point Theorem . 104 3 Cohomology 107 3.1 Singular Cohomology . 107 3.2 Exact Sequences and Functors . 109 3.3 Tor and Ext . 116 3.4 Universal Coefficient Theorems . 123 3.5 Properties of Cohomology . 126 3.6 de Rham's Theorem . 128 4 Products in Homology and Cohomology 135 4.1 Acyclic Models . 135 4.2 The K¨unneth Theorem . 136 4.3 The Cup Product . 139 4.4 The Cap Product . 144 5 Duality 148 5.1 Direct Limits . 148 5.2 The Orientation Bundle . 149 5.3 Cechˇ Cohomology . 158 i Contents Contents 5.4 Poincar´eDuality . 161 5.5 Duality of Manifolds with Boundary . 169 6 Intersection Theory 174 6.1 The Thom Isomorphism Theorem . 174 6.2 Euler Class . 178 6.3 The Gysin Sequence . 182 6.4 Stiefel Manifolds . 183 6.5 Steenrod Squares . 185 7 Higher Homotopy Theory 190 7.1 Fibration . 191 7.2 Fibration Sequences . 196 7.3 Hurewicz Homomorphisms . 199 7.4 Obstruction Theory . 200 7.5 Hopf's Theorem . 203 7.6 Eilenberg-Maclane Spaces . 205 ii 0 Introduction 0 Introduction These notes are taken from a year-long course in algebraic topology taught by Dr. Thomas Mark at the University of Virginia in the spring and fall of 2016. The main topics covered are homotopy theory, homology and cohomology, including: Homotopy and the fundamental group Covering spaces and covering transformations Universal covering spaces and the `Galois group' of a space Graphs and subgroups of free groups The fundamental group of surfaces Chain complexes Simplicial and singular homology Exact sequences and excision Cellular homology Cohomology theory Cup and cap products Poincar´eduality. The companion text for the course is Bredon's Topology and Geometry. 0.1 Differential Forms As a motivation for the study of homology in algebraic topology, we begin by discussing differential forms with the goal of constructing de Rham cohomology. This material provides a great case study which one can return to in later sections to apply results in the general homology theory. Definition. If V is a finite dimensional real vector space, the space Vp V ∗ is called the pth exterior algebra of V . The elements of Vp V ∗ are called exterior p-forms. V0 ∗ V1 ∗ ∗ Example 0.1.1. For p = 0, V = R, the underlying field. For p = 1, V = V , the ∗ vector space dual to V , which one may recall is defined as V = HomR(V; R). For p ≥ 2, Vp V ∗ may be viewed as the space of functions ! : V × · · · × V −! R | {z } p that satisfy the following properties: 1 0.1 Differential Forms 0 Introduction (a) (Multilinearity) !(: : : ; v + cw; : : :) = !(: : : ; v; : : :) + c!(: : : ; w; : : :). (b) (Alternating) !(:::;w;v;:::) = −!(:::;v;w;:::). The alternating property may be written in a more general form: jσj !(vσ(1); : : : ; vσ(p)) = (−1) !(v1; : : : ; vp); where σ is a permutation on p symbols and jσj denotes its sign (−1 if odd and +1 if even). The alternating property further implies that !(v; v; : : :) = 0, i.e. repeated terms produce 0 when an exterior form is applied to a vector. Example 0.1.2. If dim V = n and a basis of V is given, we can view an n-form ! 2 Vn V ∗ in terms of a determinant: !(v1; : : : ; vn) = det(v1 j · · · j vn): Definition. If ! 2 Vp V ∗ and η 2 Vq V ∗, their exterior product (or wedge product) is an exterior form ! ^ η 2 Vp+q V ∗ defined by X jσj (! ^ η)(v1; : : : ; vp+q) = (−1) !(vσ(1); : : : ; vσ(p))η(vσ(p+1); : : : ; vσ(p+q)); σ where the sum is over all (p; q)-shuffles σ, i.e. permutations satisfying σ(1) < : : : < σ(p) and σ(p + 1) < : : : < σ(p + q). Remark. One can alternatively write the wedge product formula as 1 X (! ^ η)(v ; : : : ; v ) = (−1)jσj!(v ; : : : ; v )η(v ; : : : ; v ): 1 p+q p!q! σ(1) σ(p) σ(p+1) σ(p+q) σ2Sp+q The wedge product defines an associative multiplication on the vector space ^ M ^ ∗V ∗ = pV ∗; p≥0 called the exterior algebra of V . n Example 0.1.3. Take some 1-forms !1;:::;!p on R and vectors v1; : : : ; vp. We claim that (!1 ^ · · · ^ !p)(v1; : : : ; vp) = det(!i(vj)): Recall by the Leibniz rule for determinants that if B = (bij) is a p × p matrix then the formula for its determinant is X det B = sgn(σ) · bσ(1)1 ··· bσ(p)p σ2Sp where Sp is the symmetric group on p symbols. Accordingly, the right hand side of the equation in the problem reads X det[!i(vj)] = sgn(σ) · !σ(1)(v1) ··· !σ(n)(vn): σ2Sp 2 0.1 Differential Forms 0 Introduction By definition of the wedge product, X (!1 ^ · · · ^ !p)(v1; : : : ; vp) = sgn(σ) · !1(vσ(1)) ··· !p(vσ(p)): σ2Sp Now this looks like we have the indices irreparably reversed, but in fact this formula is equal to det[!j(vi)], the determinant of the transpose of [!i(vj)], which we know from linear algebra is equal to the determinant of [!i(vj)] anyways. So the formula holds. Example 0.1.4. If fe1; : : : ; eng is a basis of V , there are corresponding linear functionals dx1; : : : ; dxn : V ! R uniquely defined by dxi(ej) = δij, where δij is the Kronecker delta ∗ V1 ∗ symbol. The dxi are in fact the dual basis to the ei, that is dxi 2 V = V , meaning the dxi are examples of exterior 1-forms. The wedge of two 1-forms looks like (dxi ^ dxj)(v; w) = dxi(v)dxj(w) − dxi(w)dxj(v) = viwj − vjwi; 3 where v = (v1; : : : ; vn) and w = (w1; : : : ; wn). For example, in R the standard basis gives 3 linear functionals dx; dy and dz, which act on 3-vectors in the following way: 0 1 0 1 0 1 v1 v1 v1 dx @v2A = v1 dy @v2A = v2 dz @v2A = v3: v3 v3 v3 Lemma 0.1.5. If !1;:::!p are 1-forms on V and v1; : : : ; vp 2 V then (!1 ^ · · · ^ !p)(v1; : : : ; vp) = det(!j(vi))ij: Proof. For exercise. ∗ Vp ∗ Lemma 0.1.6. If !1;:::;!n is a basis for V then a basis for V is given by f!i1 ^ · · · ^ !ip j i1 < : : : < ipg: Proof. Let fv1; : : : ; vng be the dual basis to f!1;:::;!ng, i.e. !i(vj) = δij for all i; j. Given ! 2 Vp V ∗, define the real number !i1;:::;ip = !(vi1 ; : : : ; vip ) for all increasing sequences i1 < : : : < ip. We claim that X !(vj1 ; : : : ; vjp ) = (!i1;:::;ip · !i1 ^ · · · ^ !ip )(vj1 ; : : : ; vjp ): i1<:::<ip Indeed, (!i1 ^ · · · ^ !ip )(vj1 ; : : : ; vjp ) = 0 unless fi1; : : : ; ipg = fj1; : : : ; jpg, in which case the result is 1. This proves the claimed formula, and in particular the formula holds for all size p subsets of fv1; : : : ; vng so X ! = !i1;:::;ip · !i1 ^ · · · ^ !ip : i1<:::<ip Vp ∗ P Hence the given set spans V . Now suppose ai1;:::;ip · !i1 ^ · · · ^ !ip = 0. Evaluating this on (v1; : : : ; vp) shows that all ai1;:::;ip = 0. This proves linear independence. 3 0.1 Differential Forms 0 Introduction Vp ∗ n Vp ∗ Corollary 0.1.7. If dim V = n then dim V = p . In particular, V = 0 whenever p > dim V . Lemma 0.1.8. If ! 2 Vp V ∗ and η 2 Vq V ∗ then ! ^ η = (−1)pqη ^ !. Example 0.1.9. On R3, exterior p-forms have the following forms: p = 0: ! = a 2 R. p = 1: ! = a dx + b dy + c dz where a; b; c 2 R. p = 2: ! = a dy ^ dz + b dx ^ dz + c dx ^ dy for a; b; c 2 R. Applying this to a vector may be evaluated using a determinant: 0 a b c 1 !(v; w) = det @v1 v2 v3 A w1 w2 w3 p = 3: ! = a dx ^ dy ^ dz for a 2 R. p ≥ 4: all further exterior forms are 0. The main use of exterior forms is in defining the following generalization of multivariable calculus. Let M be a smooth manifold and view V = TxM as the tangent space at some point x 2 M. Definition. A differential p-form on M is a smooth assignment of an exterior p-form Vp ∗ !x 2 Tx M to each x 2 M. n n ∼ n n When M = R , there is a natural identification TxR = R . A differential p-form on R can be written X ! = !i1;:::;ip · dxi1 ^ · · · ^ dxip i1<:::ip n 3 where each !i1;:::;ip is a smooth function R ! R. An example of a differential 1-form on R 2 3 x is x y dx + sin z dy + tan y dz. For a general n-manifold M, choose local coordinates fx1; : : : ; xng which determine a n @ @ o basis ;:::; for each tangent space TxM. This also determines a dual basis for @x1 @xn ∗ Tx M which we will write as fdx1; : : : ; dxng.
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