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Algebraic Topology Iv Epiphany Term Lecture ALGEBRAIC TOPOLOGY IV k EPIPHANY TERM LECTURE NOTES MARK POWELL This course is about cohomology of spaces, products on cohomology, and man- ifolds. Cohomology is useful primarily because it can be made into a ring using the cup product. We will study the cohomology ring and apply it to manifolds, especially of dimension 2, 3 and 4. For manifolds homology and cohomology can be related to each other in two different ways, using Poincar´eduality and using the universal coefficient theorem. The interplay between these two relations can be extremely powerful. Contents 1. CW complexes and manifolds 2 1.1. CW complexes 2 1.2. Manifolds 2 1.3. Smooth manifolds 3 1.4. Manifolds with boundary 4 2. Homology groups of manifolds 4 2.1. Recap of singular and cellular homology 4 2.2. Top dimensional homology of a manifold 6 2.3. Fundamental classes and degrees of maps 7 3. Cohomology 8 3.1. Hom groups 8 3.2. Algebra of cochain complexes 9 3.3. Cohomology of a topological space 11 3.4. CW cohomology 12 3.5. Understanding cohomology 13 3.6. Properties of cohomology 15 4. Relationships between homology and cohomology 17 4.1. Universal coefficient theorem 17 4.2. Poincar´eduality 18 4.3. An application of both 19 5. Universal coefficient theorem 19 5.1. Ext groups 19 5.2. Examples and properties 23 5.3. The universal coefficient theorem for cohomology 25 6. Univeral coefficients for homology and the K¨unneththeorem 27 6.1. Tensor products 27 6.2. Tor groups 28 1 ALGEBRAIC TOPOLOGY IV k EPIPHANY TERM LECTURE NOTES 2 6.3. Universal coefficient theorem for homology 28 6.4. The K¨unneththeorem for cohomology 29 7. Cup products 30 7.1. Definition of cup products 30 7.2. An example 32 7.3. The cohomology ring 34 7.4. More examples 36 8. Cap products 40 8.1. Definition of cap products 40 8.2. An example 42 8.3. Properties of cap product 43 9. Applications of cup and cup products combined with Poincar´eduality 44 9.1. Degree one maps 45 9.2. Computations of cup products 46 10. Proof of F2 coefficient Poincar´eduality 49 References 52 1. CW complexes and manifolds 1.1. CW complexes. Here is a better definition of CW complexes. It is important that the decomposition into cells is remembered as part of the data of a CW complex. Merely defining it as a space that \was built in a certain way" loses too much data that is needed when working with these spaces. Definition 1.1 (Finite dimensional CW complex). (a) A CW complex X of dimension −1 is ;. Its topological realisation jXj is also the empty set. (b) Inductively define higher CW complexes. A CW complex of dimension ≤ n, Xn, consists of (i) A CW complex Xn−1 of dimension ≤ n − 1, with topological realisation jXn−1j. n−1 n−1 (ii) A set of maps f'i : S ! jX jgi2In , the attaching maps. The topological realisation jXnj of X is G jXnj := jXn−1 t Dn= ∼ i2In n−1 n n−1 where for all i 2 In, x 2 S = @Di , we identify x ∼ 'i(x) 2 jX j. (c) A (finite dimensional) CW structure on a topological space Y is a CW complex X and a homeomorphism f : jXj ! Y . n n n−1 n 'i n (d) The map 'i : D ! jX j such that S ! D −!jX j equals 'i is called a characteristic map. 1.2. Manifolds. An n-dimensional topological manifold M is a topological space that is Hausdorff, second countable (countable basis of open sets), and locally ALGEBRAIC TOPOLOGY IV k EPIPHANY TERM LECTURE NOTES 3 n homeomorphic to R . That is, for all x 2 M, there is an open set Ux containing x n n and a homeomorphism 'x : Ux ! V ⊂ R where V is an open subset of R . Example 1.2. n n+1 (i) The sphere S = fx 2 R j kxk = 1g. (ii) Products of spheres Sn × Sm, of dimension n + m. (iii) The n-torus T n = S1 × · · · × S1, the product of n copies of the circle S1. n n+1 (iv) Real projective space RP = R rf0g= ∼, where x ∼ λx for all λ 2 Rrf0g. n n+1 (v) Similarly complex projective space CP = C rf0g= ∼, where z ∼ λz for all λ 2 Crf0g. (vi) The surface Σg of genus g is a 2-dimensional manifold. (vii) The Klein bottle K. n n (viii) Quotients of S or R by a free, proper, continuous action of a group are manifolds. (ix) Products of manifolds are manifolds. For this last item, we will provide several examples of manifolds obtained as orbits spaces of free actions. First, the definitions of these adjectives associated with group actions. (a) An action of a group G on a manifold X is free if g·x = x implies that g = e 2 G for every x 2 X. (b) An action of a group G on a manifold X is proper if for all x; y 2 X, there exist open sets U 3 x and V 3 y such that fg 2 G j gU \ V 6= ;g is finite. (c) A group action is continuous if the map x 7! g · x is a continuous map X ! X for all g 2 G. Now the promised examples of manifolds defined by quotients by group actions. Example 1.3. ∼ 1 (1) The group Z acts on R by n · x := x + n. The quotient is R=Z = S . 2 2 (2) The group Z acts on R by (n; m) · (x; y) = (x + n; y + m). The quotient 2 2 ∼ 2 R =Z = T . n n ∼ n (3) The group Z=2 acts on S by 1 · x = −x. The quotient S =(Z=2) = RP . 1 2n+1 2n+1 n+1 (4) The group S acts on S as follows. Consider S ⊂ C as the 2 2 2πiθ 2πiθ set of points with jz0j + · · · jznj = 1. Then e · (z0; : : : ; zn) := (e · 2πiθ 2n+1 1 ∼ n z0; : : : ; e · zn). The quotient is S =S = CP . 3 (5) The group Z=p acts on S as follows. Let p and q be coprime positive 3 2 2 2 ∼ integers. Consider S ⊂ C with jzj + jwj = 1. Write Z=p = Cp where Cp is the cyclic group generated by ξ = e2πi=p 2 S1. Then ξj · (z; w) := (ξ · z; ξq · w): 3 The quotient is S =Cp = L(p; q), the lens space. ALGEBRAIC TOPOLOGY IV k EPIPHANY TERM LECTURE NOTES 4 1.3. Smooth manifolds. All the manifolds we talk about will be smooth. n Definition 1.4. Let U ⊆ R be open. A function m f = (f1; : : : ; fn): U ! R is smooth is all partial derivatives of fi exist for all i. Definition 1.5. (i) An n-dimensional chart for a manifold M at x 2 M is a homeomorphism n Φ: U ! V where U 3 x is an open neighbourhood of x, and V ⊂ R is open. (ii) An n-dimensional atlas for M is a collection of n-dimensional charts fΦi : Ui ! S Vigi2I with i Ui = M, where I is some indexing set. (iii) An atlas fΦi : Ui ! Vigi2I is smooth if the transition map −1 Φj ◦ Φi :Φi(Ui \ Uj) ! Φj(Ui \ Uj) n is smooth as a map between open subsets of R . Definition 1.6. A smooth manifold is a topological manifold M with a choice of smooth atlas. When we say \manifold" we will usually implicitly mean smooth manifold. 1.4. Manifolds with boundary. An n-dimensional topological manifold with boundary M is a topological space that is Hausdorff, second countable, and locally n homeomorphic to either R or to the half space n n H := fx 2 R j x1 ≥ 0g: We write n @M := fx 2 M with Ux = H g: n For example, D , or a surface Σg;1. We will mostly consider closed manifolds. By definition a closed manifold is a compact manifold with empty boundary @M = ;. 2. Homology groups of manifolds We start with a recollection of homology groups, then we discuss homology of manifolds. 2.1. Recap of singular and cellular homology. Homology theories are covari- ant functors Hi : fTop spacesg ! fAbelian groupsg for each i, satisfying some axioms: homotopy invariance, long exact sequence of a pair, excision, and taking a fixed value on a point: H0(pt) = Z. ALGEBRAIC TOPOLOGY IV k EPIPHANY TERM LECTURE NOTES 5 2.1.1. Singular homology. Singular homology sends X 7! Hi(X; Z) or Hi(X) for short, defined as follows. The singular chain complex n i o Ci(X) := Z singular simplices ∆ ! X is the free abelian group generated by the singular simplices. Recall that ∆i is the i-simplex i i+1 X f(t0; : : : ; ti) 2 R j 0 ≤ ti ≤ 1; ti = 1g: i=0 The vertices tj = 1 are naturally ordered by We denote the jth face inclusion i−1 i respecting the ordering of vertices by ιj : ∆ ! ∆ . Then the boundary map @i : Ci(X) ! Ci−1(X) is defined by i i X j+1 i−1 @i(σ : ∆ ! X) = (−1) (σ ◦ ιj : ∆ ! X): j=0 The cycles are Zi(X) := ker(@i : Ci(X) ! Ci−1(X)) and the boundaries are Bi(X) := im(@i+1 : Ci+1(X) ! Ci(X)): The ith singular homology of X is Zi(X) Hi(X) := : Bi(X) 2.1.2. Cellular homology.
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