Appalachian State University. Cohomology Theories

Transcription

1 Appalachian State University Department of Mathematics Matthew Cavallo A Survey of Cohomology Theories c 2016

2 A Directed Research Paper in Partial Fulfillment of the Requirements for the Degree of Master of Arts May 2016 Approved: Dr. William J. Cook Member

3 Contents Introduction 1 1 Background Information 2 2 Building Homology Singular Homology Simplicial Homology Cellular Homology Interpretations and Homotopy Computing Homology Groups Homology Groups of S Homology Groups of the Torus Homology Groups of the Klein Bottle Homology Groups of S n Cohomology in Topology Cohomology from Homology Theories De Rham Cohomology Sheaf Cohomology Cech Cohomology Cohomology in Algebra Group Cohomology Eilenberg-Steenrod Axioms 38 Bibliography 39

4 Introduction Why are a coffee mug and a donut equivalent? The answer to this quintessential question that topologists are often asked is relatively simple: because they each have one hole. Although that may be quite apparent visually, in general, determining the number and dimension of holes in a topological space is the heart of homology theory. Homology is one of the main theories of algebraic topology, a branch of mathematics has its roots in the mid-20th century. Algebraic topology uses tools and methods of abstract algebra to answer questions about topological spaces that would difficult, if not impossible otherwise. Homology allows a group structure to be associated with a topological space, and thus information about the homology groups can be used to classify the space. The first half of this paper will construct three common homology theories (singular, simplicial, and cellular) and discuss the strengths and weaknesses of each. Afterwards, computations will be done to find the homology groups of some basic topological spaces, such as the circle S 1 and torus. The remainder of this paper will discuss homology s dual theory, cohomology. Cohomology has grown to be a quite popular area of study in the latter half of the twentieth century, with many cohomology theories arising. The biggest benefit of cohomology is that it allows a topological space to be equipped with a ring structure as well as a group structure, and thus more information about that space can be found. Several cohomology theories will be explored, including de Rham, sheaf, and Cech cohomologies. Techniques and methods of cohomology can also be applied to algebraic objects in addition to topological spaces. The last section of this paper will examine group cohomology, which uses cohomology to answer difficult questions about groups. 1

5 Chapter 1 Background Information This section will serve as a collection of background information and definitions whose use arises later in the paper. We begin with some brief category theory. Definition. A category C consists of three parts. The first is a class of objects, obj C, the next is a set of morphisms Hom(A, B) for each pair of objects A, B obj C. Lastly, we have composition. For f Hom(A, B), g Hom(B, C), we have a map Hom(A, B) Hom(B, C) Hom(A, C), denoted by (f, g) g f for every A, B, C obj C which satisfies the following axioms: (i) the family of Hom(A, B) s is pairwise disjoint; (ii) composition is associative when defined; (iii) for each A obj C, there exists an identity 1 A Hom(A, A) satisfying 1 A f = f for every f Hom(B, A), all B obj C, and g 1 A = g for every g Hom(A, C), all C obj C. Every well known area of mathematical studies is ripe with examples of categories. Groups, abelian groups, sets, and topological spaces are all examples of categories. There is even a category of categories (with a careful set up of course)! Definition. If A and C are categories, then a (covariant) functor T : A C is a pair of associations with the following properties (i) A obj A implies T A obj C (ii) if f : A A is a morphism in A, then T f : T A T A is a morphism in C, such that (iii) if f, g are morphisms in A for which g f is defined, then T (g f) = (T g) (T f) (iv) T (1 A ) = 1 T A for every A obj A. 2

6 If T f : T A T A and T (g f) = T (f) T (g), we call T a contravariant functor. Notice that contravariant functors flip arrows around. Definition. The free abelian group on a set X is the collection of all formal integer linear combinations of elements of X. Given a set X, the free abelian group on X can be denoted: { } F (X) = n 1 a n l a l l 0 ; nj Z, a j X for j = 1,..., l. We now state the Universal Property for free abelian groups. Theorem. Given a function f : X G where G is an abelian group, then there exists a unique homomorphism ˆf : F (X) G such that ˆf(x) = f(x) for all x X. The following diagram demonstrates this theorem: F (X) i X! ˆf f G (See for example [Dummit-Foote] section 10.3.) Let G be an abelian group generated by S. This means that for all g G, there exists s 1,..., s l S, n 1,..., n l Z such that g = n 1 s n l s l. This can be denoted by G = S. Then S G can be lifted uniquely to a homomorphism ϕ : F (S) G, where ϕ is surjective because S is a generator set, and so F (S) ker(ϕ) = G. If ker(ϕ) = R, we call S a set of generators and R a set of relations and G = F (S) = S R. Note that if R =, then S = F (S). ker(ϕ) Let s take a look at some examples. Example. If S = {x} and R =, then {x} = Z. If S = {x, y} and R = {5x, y}, then this would mean that 5x = 0 and y = 0. So S R = Z 5. We say a group G is finitely presented if G = S R where S and R are finite sets. Every R finitely generated abelian group is finitely presented, so G = x 1,..., x n for some x 1,..., x n { } n and R = a ij x j i = 1,..., m, a ij Z. We have j=1 a 11 x 1 + +a 1n x n = 0. a m1 x 1 + +a mn x n = 0 in G because we are modding out elements of R.. 3

7 We call A = [a ij ] the relation matrix. We can apply elementary row and column operations (swapping, scaling by ±1, adding multiples of rows/columns to rows/columns) to the relation matrix. This yields a new relation matrix whose corresponding group is isomorphic to G. Theorem. Every integer matrix can be put into a matrix of the form: d d l 0 where d 1 d2... dl (i.e. each d i successively divides the next d i ) using only elementary row and column operations. (See for example [Dummit-Foote] chapter 12.) If we obtain a matrix of this form, then we have {x 1,..., x n } generators and d 1 x 1 = 0, d 2, x 2 = 0,... relations. Thus, G = Z d1 Z dl Z n l. As will be demonstrated later, this process allows us to determine the groups appearing in homology and cohomology theories. Let s take a look at an example. Example. Let G = a, b, c 2a + 6b, a + b + c. Then we have the relations 2a + 6b = 0 and a + b + c = 0 which yield the relation matrix A: [ ] [ ] [ ] [ ] A = [ ] [ ] [ ] [ ] Therefore, G = Z 1 Z 2 Z = Z 2 Z. We will use as similar process to find homology and cohomology groups. Let F be a division ring and let n 0. Define an equivalence relation on F n+1 {0} by x y if there exists λ F {0} with x = λy. Here, F n+1 is the vector space over F consisting of all (n+1)-tuples x = (x 0, x 1,..., x n ) with coordinates x i in F. Then the quotient set (F n+1 {0})/ (or the set of all equivalence classes) is called the F -projective n-space and is denoted by F P n. Note that from this definition, we can define the real projective n-space RP n for all n 0. We also have the complex projective n-space CP n and the quaternionic projective n-space HP n. When n = 2, RP 2 is called the projective plane. Definition. A chain complex is a sequence of abelian groups S n and homomorphisms n : S n S n 1 such that n n+1 = 0 for each n Z. We define (S, ) to be a subcomplex of (S, ) if each S n is a subgroup of S n and if each n = n S n (the restriction of n to S n is n). 4

8 Definition. Let S = (S n, n ) and T = (T n, δ n ) be complexes. Then a morphism of complexes is a sequence of homomorphisms ϕ n : S n T n such that the following diagram commutes for all n: S n+1 n+1 S n ϕ n+1 ϕ n T n+1 n+1 T n Revisiting our definition of a subcomplex, we can now define a quotient. If (S, ) is a subcomplex of (S, ), then the quotient complex is S n /S n n Sn 1 /S n 1 where n : s n + S n n (s n ) + S n 1. Also note that n is well defined because n (S n) S n 1. Definition. A sequence of abelian groups and homomorphisms f n+1 f n A n+1 A n An 1 is called exact if at each A n, im f n+1 = ker f n for all n Z. A short exact sequence is an exact sequence of the form 0 A i B p C 0 Thus, a short exact sequence of complexes is an exact sequence of the form 0 S i p S S 0 with 0 denoting the zero complex. We can see from this definition that short exact sequences actually encodes the first isomorphism theorem. The sequence 0 A i B p C 0 is exact at A, if i is an a injection. It is exact at C if p is a surjection. This happens because im(0) = 0 = ker(i) and thus im(p) = ker(0) = C. Thus saying that the sequence is exact at B is the same as saying B i(a) = B ker(p) = C. Essentially, B/A = C. 5

9 Definition. A split exact sequence is an exact sequence of the form 0 A i B p C 0 for which there exists a homomorphism s : C B with p s = 1 C. From this definition, we have the following three equivalent statements: (i) The exact sequence 0 A i B p C 0 is split (ii) A and C are direct summands of B; that is, there exists a subgroup C of B with C = C via p C and im i = A because i is an injection. Therefore, B = im i C (iii) There exists a homomorphism q : B A with q i = 1 A. 6

10 Chapter 2 Building Homology We begin our discussion with the construction of the quintessential homology theory: singular homology. As will be shown, singular homology allows a means of computing homology groups without regards to the topological properties of the space. Thus, it serves as a basis for which to compare all subsequent homology theories. 2.1 Singular Homology We will now begin to define singular homology. This construction will follow closely with the construction presented in Joseph Rotman s Introduction to Algebraic Topology [Rotman 1988]. Before proceeding, we examine the following definitions. We first present a slightly alternative definition of a free abelian group. Definition. Let B be a subset of an (additive) abelian group F. Then F is free abelian with basis B if the cyclic subgroup b is infinite cyclic for each b B and F = b. The most common example of a free abelian group is the integers Z. In other words, we can say that Z is isomorphic to a free abelian group with one generator x. Similarly, Z 2 = F {x, y} = {mx + ny m, n Z}, where, for example, 3x 2y = x + x + x + ( y) + ( y). b B Definition. We define the standard n-simplex as n = { (x 1, x 2,..., x n+1 ) R n+1 each xi 0 and } x i = 1. This definition allows us to note that 0 is simply a point, 1 is a line segment, 2 is a solid triangle, and 3 is a solid tetrahedron. Definition. Let X be a topological space. A (singular) n-simplex in X is a continuous map σ : n X, where n is the standard n-simplex. For example, a 0-simplex is just a point in 7

11 X, while a 1-simplex is a path in X. Definition. Let X be a topological space. For each n 0, define S n (X) as the free abelian group with basis all singular n-simplexes in X. The elements of this free abelian group are known as (singular) n-chains. We will define S 1 (X) = 0, the trivial group. If σ : n X is continuous and n > 0, then its boundary is defined as n σ = n ( 1) i σε n i S n 1 (X) i=0 with ε n i : n 1 n being the i-th face map. If n = 0, we define 0 σ = 0. Intuitively, the i-th face map maps a point ( 0 ) onto a line segment ( 1 ), a line segment onto a triangle ( 2 ), etc. Essentially, ε 1 0 picks off the beginning of 1 and ε 1 1 picks off the end of 1, while ε 2 0, ε2 1, ε2 2 pick off the sides of the solid triangle 2. For each n 0, there is a unique homomorphism δ n : S n (X) S n 1 (X) with δ n σ = n ( 1) i ε n i i=0 for every singular n-simplex σ in X. These homomorphisms are called boundary operators. We now have a sequence of free abelian groups and homomorphisms. S n (X) n S n 1 (X) S 1 (X) 1 S 0 (X) 0 0 This is called the singular complex of X (denoted S (X)) Theorem. For all n 0, we have that n n+1 = 0. For notational convenience, we will use 2 = 0 from here on. We are now ready to formally define singular homology. Z n (X) = ker n is called the group of (singular) n-cycles. B n (X) = im n+1 is called the group of (singular) n-boundaries. Because 2 = 0, we have B n (X) Z n (X), so we can now define the n-th (singular) homology group of a space X as H n (X) = Z n(x) B n (X) = ker n im n+1 for all n 0. Before proceeding, we will examine a foundational result of homology that arises from the prior definitions. For notational purposes. we will let X denote the interior of a space X. 8

12 Theorem. (Mayer-Vietoris) If X 1, X 2 are subspaces of X with X = X 1 X 2, then there is an exact sequence H n (X 1 X 2 ) (i 1,i 2 ) H n (X 1 ) H n (X 2 ) g j H n (X) D H n 1 (X 1 X 2 ) with i 1, i 2, g, j inclusions and D = dh 1 q, where h, q are inclusions and d is the connecting homomorphism of the pair (X 1, X 1 X 2 ) (see [Rotman 1988] page 107). space. As the the reader can see, we have defined the n-th homology group for any topological A benefit of using singular homology is the fact that we were able to compute the homology groups without considering any properties of the topological space X. However, such computations prove to be rather difficult, if not impossible, due to the size of the group of n-chains. We will now shift our focus to spaces that will allow easier computations and begin our discussion on simplicial homology. 2.2 Simplicial Homology Simplicial homology is much easier to use and lends itself nicely to computating homology groups. As we will see, simplicial homology can be used for any triangularizable topological space. Perhaps unsurprisingly, many well known mathematical objects such as the circle and torus are triangularizable. We begin our disussion with some preliminary definitions about simplicial complexes. Definition. The vertex set of a q-simplex s = [v 0,..., v q ] is denoted Vert(s)= {v 0,..., v q }. If s is a simplex, then a face of s is a simplex s with Vert(s ) Vert(s). There are faces in different dimensions. For example, a 0-face is simply the vertices of a simplex, a 1-face the edges, etc. Definition. A finite simplicial complex K is a finite collection of simplexes in some euclidean space such that: (i) if s K, then every face of s also belongs to K (ii) if s, t K, then s t is either empty or a common face of s and of t. The dimension of K is the largest integer n such that K contains an n-simplex Let s look at a few examples. Example. Consider the following diagram. 9

13 D B C A Figure: A simplicial complex Although this may not seem like a simplicial complex, it satisfies conditions (i) and (ii), as the the intersection B is a face of both the triangle and the line segment. Example. The following is not a simplicial complex. b e S d T f a Figure: Not a simplicial complex c It is easy to see that the figure above is not a simplicial complex. Although S and T themselves are, S T = d, which is a face of T but not a face of S. Definition. If K is a simplicial complex, its underlying space K is the subspace (of the ambient Euclidean Space) is: K = s Definition. A topological space X is a polyhedron if there exists a simplicial complex K and a homeomorphism h : K X. The ordered pair (K, h) is called a triangulation of X. s K Definition. An oriented simplicial complex K is a simplicial complex and a partial order on Vert(K ) whose restriction to the vertices of any simplex in K is a linear order. For example, we can order the vertices of the following triangle A < B < C: 10

14 B A C Figure: The linear ordering A < B < C orients this triangle. Definition. An abstract simplicial complex K is a family of nonempty subsets of V, called simplexes, such that (i) if v V, then {v} K (ii) if s K and s s, then s K. Definition. If K is an oriented simplicial complex and q 0, let C q (K) be the abelian group having the following presentation: 1. Generators: all (q + 1)-tuples (p 0,...p q ) with p i Vert(K) such that {p 0,..., p q } spans a simplex in K. 2. Relations: (i) (p 0,...p q ) = 0 if some vertex is repeated (ii) (p 0,...p q ) = (sgn π)(p π0, p π1,..., p πq ), where π is a permutation of {0, 1,..., q}. So if dimension is n, then C m (K) = 0 for m > n because we must have repeated vertices. We will now begin constructing simplicial homology. Definition. We define q : C q (K) C q 1 (K) as follows: q ( (p 0,..., p q ) ) = q ( 1) i p 0,..., ˆp i,..., p q i=0 where (p 0,..., p q ) is a generator and ˆp i tells us to delete the vertex p i. Then extend q linearly to all C q (K). Theorem. If K is an oriented simplicial complex of dimension m, then 0 0 C m (K) C 1 (K) C 0 (K) 0 11

15 is a chain complex. Note that as with singular homology we have 2 = 0. If K is an oriented simplicial complex, then Z q (K) = ker q is called the group of simplicial q-cycles. B q (K) = im q+1 is called the group of simplicial q-boundaries. H q (K) = Z q(k) B q (K) is the q-th simplicial homology group. As with singular homology, simplicial homology has both upsides and downsides. The biggest benefit of using simplicial homology is that allows for relatively easy calculations of homology groups of spaces. This will be demonstrated later. Unfortunately, when using simplicial homology, we are restricted to spaces that can be triangulated. However, many interesting spaces fall into this category. For example, J.H.C. Whitehead proved that every smooth manifold can be triangulated [Whitehead]. 2.3 Cellular Homology We now shift our attention from triangles to bubbles. At its core, cellular homology involves taking the singular homology of topological spaces built out of n-dimensional cells (which can be thought of as bubbles). This qualification turns out to be a happy medium between the amount of spaces used in simplicial (triangularizable spaces) and singular homology (all spaces). We now begin our construction of cellular homology. However, we must first define several new terms before we are prepared to do so. Definition. An n-cell e n is a homeomorphic copy of the open n-disk D n S n 1. For example, e 1 is an open interval, e 2 is an open disk, etc. We say an n-cell has dimension n if dim(e n ) = n. Definition. A Hausdorff space X is locally compact if, for each x X and every open set U containing x, there exists an open set W with W compact and x W W U. Let X be a set covered by subsets A j, where j lies in some (possibly infinte) index set J. So if X = A j, then assume: j J (i) each A j is a topological space (ii) for each j, k J, the topologies of A j and A k agree on A j A k. (iii) for each j, k J, the intersection A j A k is closed in A j and in A k. Then the weak topology on X determined by {A j : j J} is the topology whose closed sets are those subsets F for which F A j is closed in A j for every j J. 12

16 Assume that a topological space X is a disjoint union of cells (i.e. X = {e : e E}). For each k 0, the k-skeleton X (k) of X is defined by X (k) = {e E Note that X (0) X (1) and X = k 0 X (k). } dim(e) k Definition. A CW complex is an oriented triple (X, E, Φ) where X is a Hausdorff space, E is a family of cells in X, and Φ = {Φ e : e E} is a family of maps, such that (i) X = {e : e E }; (ii) for each k-cell e E, the map Φ e : (D (k), S k 1 ) (e X (k 1), X (k 1) ) is a relative homeomorphism We call a continuous map g : (X, A) (Y, B) a relative homeomorphism if g : (X A) (Y B) is a homeomorphism); (X A) (iii) if e E, then its closure e is contained in a finite union of cells in E; (iv) X has the weak topology determined by {e : e E}. We are almost ready to discuss cellular homology. Our final preliminary definition is that of relative homology, which will play a critical role in our definition of cellular homology. Definition. Let A be a subspace of X. Then we can derive a short exact sequence (using the chain complex of singular chains) 0 S (A) S (X) S (X)/S (A) 0 We define the n-th relative homology group H n (X, A) as H n (S (X)/S (A)). This definition grants us the following theorem: Theorem. If A is a subspace of X, there is an exact sequence H n (A) H n (X) H n (X, A) d H n 1 (A) This result will prove very useful in our definition of cellular homology. Now that we have thoroughly defined relative homology, we begin working our way to cellular homology. A filtration of a topological space X is a sequence of subspaces {X n : n Z} with X n X n+1 for all n. A filtration is cellular if: (i) H p (X n, X n 1 ) = 0 for all p n; (ii) for every m 0 and every continuous σ : m X, there is an integer n with im σ X n. 13

17 A cellular space is a topological space with a cellular filtration. If X and Y are cellular spaces, then a cellular map is a continuous function f : X Y with f(x n ) Y n for all n Z If X is a cellular space and k 0, we define W k (X) = H k (X k, X k 1 ) We define d k : W k (X) W k 1 (X) as the composite d k = i. The following diagram displays our definition: H k (X k, X k 1 ) d k H k 1 (X k 1 ) H k 1 (X k 1, X k 2 ) i Here, i : (X k 1, ) (X k, X k 1 ) is the inclusion map and is the connecting homomorphism coming from the following long exact sequence of the pair (X k, X k 1 ). H k (X k 1 ) H k (X k ) H k (X k, X k 1 ) H k 1 (X k 1 ) If X is a cellular space, then (W (X), d) is a called the cellular chain complex of the filtration of X. It is a chain complex as d k d k+1 = 0. Since cellular spaces are built from gluing spheres together, cellular homology is essentially determined by the interactions between the homology of spheres. Using cellular homology, we are able to compute the homology groups of the real projective space for any dimension. Theorem: If n is odd, then Z if p = 0 or p = n H p (RP n ) = Z 2 if p is odd and 0 < p < n 0 otherwise If n is even, then Z if p = 0 H p (RP n ) = Z 2 if p is odd and 0 < p < n 0 otherwise As we can see, cellular homology can be very useful. Computing the homology groups of RP n would be very difficult without it. However, there is a lengthy and somewhat inaccessible plethora of background information required in both singular homology as well as basic topology to even define this theory. 14

18 2.4 Interpretations and Homotopy Now that we have defined three separate homology theories, one may be wondering what the underlying purpose is. The zeroth homology group exactly characterizes the number of path connected components of a space. In particular, H 0 (X) is isomorphic to the free abelian group on t generators where t is the number of path connected components. When n > 0, roughly speaking, the n-th homology group measures the number of n- dimensional holes in a space. For example, H 0 (R n ) = Z because R n is path connected and H m (R n ) = 0 for all m > 0 because R n has no holes. As another example, we will soon see that H m (S n ) = Z for m = 0, n and H m (S n ) = 0 otherwise. Intuitively this is because an n-sphere has a single n-dimensional hole or bubble (and has one path connected component). However, as can be seen from the homology groups of RP n, a geometric interpretation of homology groups can be a bit more complicated (what kind of hole can be represented by Z 2?). Before moving on to computing homology groups, it is important to discuss the relationship between homology and another theory in algebraic topology. We first present the definition of homotopy for functions and then for spaces: Definition. If X and Y are topological spaces and if f 0, f 1 are continuous maps from X to Y, then f 0 is homotopic to f 1 if there is a continuous map F : X I Y with F (x, 0) = f 1 (x) and F (x, 1) = f 2 (x). This is denoted by f 0 f 1. Moreover, we say that topological spaces X and Y are homotopic if there are continuous maps f : X Y and g : Y X such that g f 1 X and f g 1 Y [Munkres]. Essentially, homotopy determines if one space can be continuously deformed into another. Although this may sound similar to the idea of a homeomorphism, it is not as strict a relation. For example, a circle, cylinder (just the side without top and bottom), the deleted plane R 2 {0}, and the Möbius strip are all homotopic. However, none of these are homeomorphic to each other. Homology is a functor. If one has a continuous map f : X Y then we get an induced homomorphism f : H (X) H (Y ). Most importantly, if f g, then f = g. This implies that homology is homotopy invariant, meaning that if X can be deformed through a homotopy into Y, then H n (X) = H n (Y ) for all n. This means that if any of the n-th homology groups aren t isomorphic, then X and Y can t be homotopic. Therefore, this means X and Y are not homeomorphic either. For example, H n (S n ) = Z = 0 = Hn (R n ), and therefore we know that S n R n. However, simply because the homology groups of two spaces are isomorphic doesn t mean the spaces themselves are homotopic. This is where cohomology will be of great use, as equipping a space with a ring structure provides another tool for classifying spaces. However, there are instances when even cohomology fails to distinguish between two spaces. This shouldn t be too surprising, as a theory that never failed would be computationally impossible. 15

19 Chapter 3 Computing Homology Groups Now that we have defined three different kinds of homology, let s compute some homology groups. This is where simplicial homology is most beneficial, as many of these computations rely on being able to construct objects from the identification of sides and vertices of rectangles. 3.1 Homology Groups of S 1 We will first start with the circle S 1 Figure: S 1 In order to compute the homology groups of S 1, we will use techniques gathered from simplicial homology and make use of the well known topological fact that a circle is homeomorphic to a triangle. In order to compute the homology groups S 1, we will orient the triangle K as follows: z c b x a y Figure: A complex K which is homeomorphic to S 1. We now need to construct a chain complex. Since our triangle has dimension m = 2, we have the following chain complex. 0 C 2 (K) 2 C 1 (K) 1 C 0 (K)

20 Note that we have 0-dimensional simplexes x, y, x and 1-dimensional simplexes a, b, c. Since the interior of the triangle K is empty, there is no 2-dimensional simplex and thus C 2 = 0. Because we have three 0-dimensional simplexes, C 0 = x y z. Elements in C 0 can be expressed in the form αx + βy + γz with α, β, γ Z. Similarly, C 1 = a b c and elements can be expressed as la + mb + nc with l, m, n Z. Let s examine the boundary operators. We first note that 0 is simply the zero map and 2 simply sends 0 to 0. 1 (a) = y x 1 (b) = z y 1 (c) = x z We are now ready to compute the cycles and boundaries. But since 1 is a homomorphism, we have Z 0 = ker 0 = C 0 = x y z = x, y, z B 0 = im 1 = (la + mb + nc) 1 (la + mb + nc) = l 1 (a) + m 1 (b) + n 1 (c) = l(y x) + m(z y) + n(x z) Thus, we can express B 0 as: B 0 = y x, z y, x z Therefore, we have that H 0 (K) = Z, as H 0 is defined as Z 0 B 0. We can think of the quotient as sending elements of B 0 to 0, and so this would mean y x = 0, z y = 0, and x z = 0 (mod B 0 ). Then x = y = z. Thus, the elements in this quotient group can be expressed as nx + B 0. We now turn our attention to H 1 (K). In order to compute H 1 (K), we must find Z 1 (K) and B 1 (K). We know that Z 1 = ker 1, and we can find this kernel by setting (la + mb + nc) = 0 and solving. We know that (la + mb + nc) = l(y x) + m(z y) + n(x z), so we have (n l)x + (l m)y + (m n)z = 0, so n l = 0, l m = 0, and m n = 0. But this can only occur if l = m = n, and thus As was previously stated, 2 = 0. Therefore, we have Z 1 = ker 1 = a + b + c = Z. H 1 (K) = Z 1 B 1 = Z 0 = Z Because every higher dimension homology group will be simply be the zero group, we now state our results: 17

21 H n (S 1 ) = { Z if n = 0 or n = 1 0 otherwise 3.2 Homology Groups of the Torus The next few examples are not strictly simplicial complex computations. Instead, they are a simplified version of a complex which makes computations easier (see [Rotman 1988] chapter 7, pages ). Let s now compute the homology of another common mathematical object, the torus. We will let X be the torus arising from the following rectangle (we will call it S). v a v b b v a Figure: Torus v Here we identify the a sides together, the b sides together, and all of the vertices are identified as one. This results in a donut shaped region. We see that we have one 0-dimensional complex (the vertex v), two 1-dimensional complexes (the edges a and b), and also a 2-dimensional complex, the interior face of the rectangle S. Thus, C 0 = v, C 1 = a b, so elements in C 1 can be expressed in the form na+mb, with n, m Z. We also have C 2 = S. Similar to our previous calculation, we now have the following sequence: 0 3 C 2 (X) 2 C 1 (X) 1 C 0 (X) 0 0 Note that 3 is simply the zero map. We must now apply the boundary operators to each of these complexes. We find: 2 (S) = a + b a b = 0 1 (na + mb) = n 1 (a) + m 1 (b) = n(v v) + m(v v) = 0 0 (v) = 0 We quickly see that each boundary operator is simply the zero map. We now find the cycles and we will then be able to compute the homology of a torus. 18

22 Z 0 = ker 0 = C 0 = v = Z B 0 = im 1 = 0 Thus H 0 = Z 0 = Z Similarly, Z 1 = ker 1 = C 1 = a b = Z Z B 1 = im 2 = 0 Thus H 1 = Z Z 0 = Z Z Lastly, we have Z 2 = ker 2 = C 2 = S = Z B 2 = im 3 = 0 Thus H 2 = Z 0 = Z In summary, for some torus X, we have: Z if n = 0 or n = 2 H n (X) = Z Z if n = 1 0 otherwise 3.3 Homology Groups of the Klein Bottle We now turn our attention to another familiar mathematical object: the Klein Bottle, which can be obtained from the following rectangle: v a v b b v a Figure: Klein Bottle v 19

23 Identify sides and vertices as in the case with the torus above. As before, we construct the following complex 0 3 C 2 (X) 2 C 1 (X) 1 C 0 (X) 0 0 We see that there is one 2-dimensional complex, (the face of the rectangle) call it P. We have two 1-dimensional complexes, a and b, and one 0-dimensional complex, the vertices v. Thus C 2 = P, C 1 = a b, and C 0 = v. We now compute the boundaries of these complexes. Finally, we have (P ) = a + b + a b = 2a (a) = v v = 0 = (b) (v) = 0 Z 0 = C 0 = v = Z B 0 = 0 H 0 = Z 0 = Z Next, we have Z 1 = C 1 = a b B 1 = 2a H 1 = a b 2a = Z 2 Z This gives us the following cycles and boundaries Z 2 = 0 B 2 = 0 H 0 = 0 In summary, we have the following homology groups for the Klein Bottle: Z if n = 1 H n = Z Z 2 if n = 2 0 otherwise 20

24 3.4 Homology Groups of S n Although the circle, the torus, and the Klein bottle are very different mathematical objects, the computations were similar as they arose from a 2-dimensional triangulation. Our next computation will be to compute the homology groups of a sphere and of higher dimensional spheres. Figure: The sphere S 2. We note that the sphere is homeomorphic to a tetrahedron. Therefore, we we could compute the homology groups of the sphere in a similar manner to our other shapes using applying the boundary operator to each face. However, a simpler way exists! The sphere is simply the union of a 0-cell and a 2-cell (in fact, S n = e 0 e n ). Therefore, the homology groups of an n-dimensional sphere will be H m (S n ) = { Z if m = 0, n 0 otherwise 21

25 Chapter 4 Cohomology in Topology We have now thoroughly explored three different kinds of homology theories, examined their strengths and weaknesses, and have computed several homology groups of common mathematical objects. However, we now will examine a more powerful tool. We turn our attention from homology to its dual theory, that of cohomology. Cohomology has its roots in topology and has become a very popular area of study over the last sixty years. When applied to a topological space, cohomology equips the space with not only a group structure, but a ring structure as well. Thus, more information can be learned. Throughout the next several chapters, we will examine how cohomology can arise from homology theories. We will first examine how to dualize the construction of singular and simplicial homologies. Then we will look at de Rham cohomology followed by sheaf and Cech cohomology theories. Our final chapter on cohomology reveals another wonderful strength of the theory; the fact that it can be applied to purely algebraic objects as well as topological spaces. We will build group cohomology in two distinct fashions before discussing the first two cohomology groups and their usefulness in studying problems in group theory. 4.1 Cohomology from Homology Theories Before proceeding, we must revisit a term from our background information on category theory. If we have an abelian group G, the Hom functor Hom(, G): Ab Ab is a contravariant functor. This means (φ ψ) # f = f φ ψ = ψ # (f φ) = ψ # φ # f (so Hom functor is contravariant). In particular, the Hom functor behaves on morphisms as follows: if φ : A B is a homomorphism, then φ # : f f φ. We also know that Hom(, G) is an additive functor (i.e. (φ + ψ) # = φ # + ψ # ), and so φ # is the zero map whenever φ is. Although this definition may seem very abstract, there is actually a fairly intuitive way to 22

26 describe these groups. If F is a free abelian group with basis B, then the elements in Hom(, G) correspond to functions from B to G. Constructing cohomology will, in several cases, be as simple as applying the Hom functor to a previously discussed chain complex and then taking the homology of the new complex. Example. (Singular Cohomology) Recall our singular complex: n+1 S n (X) n S n 1 (X) S 1 (X) 1 S 0 (X) 0 0 We now apply the Hom functor to this complex and have that if (S (X), ) is the singular complex of a space X, then, for every abelian group G, 0 Hom(S 0 (X), G) # 1 Hom(S 1 (X), G) # 2 Hom(S 2 (X), G) is a complex denoted by Hom(S (X), G). Notice that because Hom is a contravariant functor, all of our arrows point in the opposite direction. We see that this is in fact a complex as # n+1 # n = ( n n+1 ) # = 0 # = 0. and in our complex, we have that Hom(S n (X), G) corresponds to functions with values from G on the n-simplexes in X If we replace # n+1 with δn, we arrive at the following cochain complex: 0 Hom(S 0 (X), G) δ0 Hom(S 1 (X), G) δ1 Hom(S 2 (X), G) We can now define singular cohomology. Definition. Let G be an abelian group and let X be a topological space. If n 0, then the group of (singular) n-cochains in X with coefficients in G is Hom(S n (X), G). The group of n-cocycles is ker δ n and is denoted by Z n (X; G). The group of n-coboundaries is im δ n 1 and is denoted by B n (X; G). As with homology, we have B n Z n because δ 2 = 0. Thus, the nth cohomology group of X with coefficients in G is H n (X; G) = Zn (X; G) B n (X; G) = ker δn im δ n 1 = ker # n+1 im n # Note that we could have defined homology with coefficients in G, but for the sake of simplicity we avoided doing so. Our homology theories have coefficients in Z. Lets look at another example. Example. (Simplicial Cohomology) Recall our simplical chain from earlier: (C (K), G) = C m+1 (K) m+1 C m (K) m C m 1 (K) We now apply the Hom functor and have a simplicial cochain complex: 23

27 Hom(C m 1 (K), G) δi 1 Hom(C m (K), G) δi Hom(C m+1 (K), G) Thus, the simplicial cohomology groups are H n (K; G) = H n (Hom(C (K), G)). [Rotman 2009] Many of our previous results from homology hold in cohomology due to the contravariant nature of the Hom functor. However, what makes cohomology of such interest is the ring structures that naturally arise in these theories. While cohomology groups can be enriched and turned into rings. The same is not true for homology groups. Definition. A ring R is a graded ring if there are addititive subgroups R n, n 0, such that: (i) R = n 0 R n (ii) R n R m R n+m for all n, m 0 We see that (i) is equivalent to saying that R is the direct sum of additive groups. Condition (ii) means that if x R n, y R m, then xy R n+m. An example of a graded ring would be the polynomial ring R = A[x], with A being a commutative ring and R n = {ax n a A}. If 0 i d, we define maps λ i, µ i : δ i δ d by λ i : (t 0,..., t i ) (t 0,..., t i, 0,..., 0) (called a front face), and µ i : (t 0,..., t i ) (0,..., 0, t 0,..., t i ) (called a back face). Notice the similarity to the ith face maps from homology, as these behave in a similar fashion. Given a space X and an abelian group G, we will write S n (X, G) = Hom(S n (X), G) and S (X, G) = S n (X, G). n 0 Definition. Let X be a space, and let R be a commutative ring. If φ S n (X, G) and θ S m (X, R), then their cup product φ θ S n+m (X, R) is defined as: (σ, φ θ) = (σλ n, φ)(σµ m, θ) for every (n + m)-simplex σ in X, with the right side of the equality being the product of two elements in the ring R. The cup product has the property that ( φ i ) ( θ j ) = φ i θ j i,j with φ i S i (X, R) and θ j S j (X, R). From this definition, we find that if X is a space and R is a commutative ring, then S (X, R) = S n (X, R) is a graded ring under the cup product. However, this ring is absolutely massive (as well as noncommutative)! The good news is that 24

28 the ring structure of S (X, R) will be passed on to H n (X; R), which will be much easier to n 0 work with. As with homology, there are many different cohomology theories. The remainder of this paper will begin to analyze some additional popular cohomology theories and assess their strengths and weaknesses. The first such theory we will examine is that of de Rham cohomology. 4.2 De Rham Cohomology De Rham cohomology was created by Georges de Rham in the late 1920s. One may find this to be the most accessible of the cohomology theories presented, as it heavily involves differential forms. Some background definitions and results will be presented before formally defining de Rham cohomology. Definition. If A is a commutative ring with 1, an A-module is an abelian group M equipped with scalar multiplication A M M. This multiplication is denoted by (a, m) am and we have the following identities for all m, m M and a, a A: (i) a(m + m ) = am + am ; (left distribution) (ii) (a + a )m = am + a m; (right distribution) (iii) (aa )m = a(a m); (associativity) (iv) 1m = m. (identity) We now have that if A is a field, then an A-module is a vector space over A. In fact, we can view the ring A as an A-module by taking scalar multiplication to be the defined multiplication of A. We also have that if M 1,..., M n are A-modules, then so is their direct sum M 1 M n. We can also take the direct sum of n copies of A, denoted A (n). This is called a free A-module. We shall denote e i A (n) as the n-tuple having 1 in the ith position and zero everywhere else (similar to a unit vector), then each element m A (n) can be uniquely represented by m = a i e i, with a i A. Definition. If M is an A-module and p 0, then the p th exterior power of M, denoted by p M, is the abeliean group wit the following presentation: Generators: A M M (p factors M.) Relations: For all a, a A, m i, m i M, we have 1. (a, m 1,..., m i + m i,..., m p) = (a, m 1,..., m i,..., m p ) + (a, m 1,..., m i,..., m p) for all i; 2. a + a, m 1,..., m p ) = (a, m 1,..., m p ) + (a, m 1,..., m p ); 25

29 3. (aa, m 1,..., m i,..., m p ) = (a, m 1,...a m i,..., m p ) for all i 4. (a, m 1,..., m p = 0 if m i = m j for some i j Definition. If X is a connected open subset of R n and if p 0, then we write Ω p (X) = p (A(X) (n) ) where A(X) are smooth functions from X to A (i.e. infinitely differentiable functions). We call elements ω Ω p (X) differential p-forms on X. Then the exterior derivative d p : Ω p (X) Ω p+1 (X) is defined as d p (ω) = d 0 (α i1...i p ) dx i1 dx ip. Example. A differential 0-form is a scalar-valued function. expressed as A differential 1-form ω can be ω = F 1 (x 1,..., x n )dx 1 + F 2 (x 1,..., x n )dx F n (x 1,..., x n )dx n where F j for j = 1,..., n is a scalar-valued function on a subset of R n. [Colley] Therefore, a connected open set X in R n determines a sequence of homomorphisms 0 Ω 0 (X) d0 Ω 1 (X) d1 dn 1 Ω n (X) 0 It is not hard to show that d 2 = 0, and thus this sequence is a complex. The homology groups of this complex are called the de Rham cohomology of X. Using de Rham cohomology (as well as the Mayer-Vietoris Theorem), we can compute the cohomology of S 1 (the unit circle). Example. Suppose U and V are open sets with U covering everything except the north pole and V covering everything except the south pole. Since U and V form an open cover of S 1, we can apply the Mayer-Vietories Therorem. We have the following exact sequence 0 H 0 (S 1 ) H 0 (U) H 0 (V ) H 0 (U V ) δ δ H 1 (S 1 ) H 1 (U) H 1 (V ) H 0 (U V ) 0 We note that U and V are homotopic, as they are simply punctured circles, which are in fact homeomorphic to R. therefore, H 0 (R) = H 0 (U) = H 0 (V ) = R. Since U V can be retracted to two points, H 0 (U V ) = R R. Since U and V are star-shaped, H 1 (U) = H 1 (V ) = 0, and since U V retracts to two points of dimension 0, H 1 (U V ) = 0 as well. We know that the image of δ is equal to the kernel of the map from H 1 (S 1 ) to H 1 (U) H 1 (V ), but H 1 (U) H 1 (V ) = 0, and so δ is surjective. The kernel of δ is equal to the image of the map from H 0 (U) H 0 (V ) to H 0 (U V ), but since this mapping is the subtracting of constant functions, the image is simply the set of constant functions, and is thus isomorphic to R. Therefore, H 1 (S 1 ) = R. [Greene] De Rham cohomology may actually be the most recognizable cohomology theory due to its use of differential forms. In fact, most advanced calculus students have worked with coboundaries 26

30 and cocycles without even realizing it! advanced calculus and the cohomology definitions we have used. There is a distinct link between the terminology in For example, a k-form ω is closed if dω = 0, while it is called exact if ω = dα for some (k 1)- form α. In terms of cohomology, closed forms are cocycles and exact forms are coboundaries. To further see the link between calculus and cohomology, consider the following complex when n = 3 (let X R 3 ): 0 Ω 0 (X) Ω 1 (X) Ω 2 (X) Ω 3 (X) 0 If ω Ω 0 (X), then ω = α(x, y, z), a C -function on X, and d 0 ω = α x + α y + α z which is a 1-form resembling α. In general, 0 and 3-forms can be identified with scalar valued functions and 1 and 2-forms can be identified with vector fields. Then using similar calculations, we can find that the derivative of 1-form will match the curl operator, while the derivative of a 2-form will match the divergence operator. d 2 = 0. The familiar results that f = 0 and ( F) = 0 are really just special cases of Also, the result from multivariate calculus that on a simply connected region X R 3, F = 0 is equivalent to F = f for some f is really just the statement that closed and exact 1-forms on X are the same thing. This in turn is just the statement H 1 (X) = 0. These results can be applied to provide the classical Stokes theorem (n = 3, p = 3), Green s theorem (n = 2, p = 1), and the fundamental theorem of calculus (n = 1, p = 0). [Rotman 1988] 4.3 Sheaf Cohomology The next cohomology theory we will explore is that of sheaf cohomology. It is a relatively newer cohomology theory (development increased greatly after 1950) and allows us to calculate information on the global sections of a sheaf. As a general idea, sheaf cohomology allows us to measure how far from being exact a sequence of sheaves (will be defined) is. As before, we will examine some background definitons and results before defining the cohomology. We will follow constructions as laid out in [Rotman 2009]. Definition. A local homeomorphism a continuous map ρ : E X between topological spaces E and X such that for each e E, there is an open neighborhood S of e (called a sheet) with ρ(s) open in X and ρ : S ρ(s) a homeomorphism. The triple (E, ρ, X) is called a S protosheaf if ρ is surjective. In this triple, we call the space E the sheaf space, ρ the projection, and X the base space, and for each x X, the fiber p 1 (x) (denoted by E x is called the stalk over x. A simple example of a protosheaf is the triple (R, ρ, S 1, with ρ = e 2πix ). We now look at some important results about protosheaves. 27

31 Theorem. Let (E, ρ, X) be a protosheaf. Then (i) The sheets form a base of open sets for E. (ii) ρ is an open map (meaning it maps open sets to open sets) (iii) Each stalk E x is discrete (iv) Let (U i ) i I be a family of open subsets of X, and let U = i I U i. If f, g : U Y for some space Y and f = g for all i I, then f = g. Ui Ui (v) Let (U i ) i I be family of open subsets of X and, for each (i, j) I I, define U (i,j) = U i Uj. If (f i : U i Y ) i I are continuous maps satisfying f i = f j for all (i, j) I I, U(i,j) U(i,j) then there exists a unique continuous map f : U Y with f = f i for all i I. Ui Definition. We define an etale-sheaf as follows. Let X and E be topological spaces. If ρ : E X is continous, then S = (E, ρ, X) is an etale-sheaf of abelian groups if (i) (E, ρ, X) is a protosheaf, (ii) the stalk E x is an abelian group for each x X, (iii) inversion and addition are continuous. Using etale-sheaves, we are able to study sections over a topological space. If S = (E, ρ, X) is an etale-sheaf of abelian groups and U X is an open set, then a section over U is a continuous map σ : U E such that ρ σ = 1 U. If U = X, then we refer to σ as a global section. We denote the sections as Γ(U, S) = {sections σ : U E} (we define Γ(, S) = {0}). We can see that sections Γ(U, S) describe the local properties of a base space X while Γ(X, S) describes the global properties. With this definition, we arrive at the following theorem: Theorem. Let S = (E, ρ, X) be an etale-sheaf of abelian groups, and let F = Γ(, S ). Then: (i) F(U) is an abelian group for each open U X. (ii) F = Γ(, S) is a presheaf of abelian groups on X (called the sheaf of sections of S), (iii) The function z : X E, defined by z(x) = 0 x E x is a global section (this is referred to as the zero section). We now begin to work our way towards the definition of a presheaf. Definition. Let X be a topological space. Then the topology U of X is actually a category. In this category, obj(u) = U)and if U, V U, then 28

32 Hom(U, V ) = { if U V {i U V } if U V with i U V : U V simply being the inclusion map. If U is the topology of some topological space X and C is a category, then a presheaf over X is a contravariant functor P : U C. We now arrive at a very important condition. A presheaf {F, ρ V U } of abelian groups on a space X saisfies the equalizer condition if (i) (Uniqueness) for every open set U and open cover U = i I U i, if σ, τ F(U) satisfy σ = τ for all i I, then σ = τ Ui Ui (ii) (Gluing) for every open set U and open cover U = i I U i, if σ i F(U i ) satisfy (Ui σ i = σ j for all i, j, then there exists a unique σ F(U) with σ Ui = σ i for Uj ) (Ui Uj ) all i I. We can now define a sheaf. Definition. A sheaf of abelian groups over a space X is a presheaf F on X that satisfies the equalizer condition (always assume F( ) = 0). We are almost ready to define sheaf cohomology, but we need a few more preliminary definitions. We will first add to the definition of a module discussed in the previous section. Definition. A module Q is an injective module if given any module A, a sub-module A, and a homomorphism A Q, there is an extension A Q. [Cartan-Eilenberg] An injective resolution of A obj(a), where A is an abelian category, is an exact sequence E= 0 A η E 0 d0 E 1 d1 E 2 in which each E n is injective. If E is an injective resolution of A, then its deleted injective resolution is the complex E A = 0 E 0 d0 E 1 d1 E 2 Therefore, if X is a topological space, then sheaf cohomology is defined as follows: for every sheaf F over X, the sheaf cohomology is H q (F) = (R q Γ)(F), where R q Γ is the q-th right derived functor of Γ. This can be thought of as taking an injective resolution E of F, deleting F to obtain E F, applying Γ, and taking homology to find: H q (F) = H q (ΓE F ). [Rotman 2009] The study of sheaves and sheaf cohomology has grown to be an area of interest in algebraic geometry. It can be used to answer the Cousin problems in complex variables. Another cohomology that involves sheaves is Cech cohomology, which we will now discuss. 29

33 4.4 Cech Cohomology Cech cohomology is named after the mathematician Eduard Cech. It involves the refinement of open covers and some results from sheaf cohomology. We begin our discussion of Cech cohomology by revisiting an earlier definition. Let V be a finite set, and recall that an abstract simplicial complex K is a family of nonempty subsets of V, called simplexes, such that (i) if v V, then {v} K (ii) if s K and s s, then s K. We now arive at the following definition. If U = (U i ) i I is an open cover of a topological space X, then the nerve N(U) is the abstract simplicial complex with vertices Vert(N(U)) = U and q-simplexes all (q + 1)-tuples σ of distinct open sets. We have σ = [U i0,..., U iq ] with each U ij disjoint from the others. We will now look at an example that will aid in our construction of Cech cohomology. Example. Let K be an abstract simplicial complex. We will denote the complex C. (K) = C q (K) q C q 1 (K) If G is an abelian group, then C q (K, G) = Hom(C q (K), G) is called the simplicial q-cochains with coefficients in G. Since C q (K) is free abelian, a q-cochain f : C q (K) G is determined by its values on the basis q (K), the family of all q-simplexes in K. So f can be viewed as a function from q (K) to G. We define the differential δq : C q (K, G) C q+1 (K) as the induced map f f q. The homology groups of the complex Hom(C. (K), G) are called the simplicial cohomology groups of K with coefficients in G. Since the nerve of an open cover U of a space X is an abstract simplicial complex, we can now define cohomology groups of N(U). If q q-cochain is a Z-linear combination of functions f : q define (δ q q+1 f)(τ) = ( 1) j f[u i0,..., Ûi j,..., U iq+1 ] j=0 is the set of all q-simplexes in N(U), then a G. Let τ be a (q + 1)-simplex and to obtain a complex of abelian groups C (N(U), G). Thus, the homology groups of the complex C (N(U), G) are called the cohomology groups of the open cover U with coefficients in G. These groups are denoted by H q (U, G). Using this construction, we can actually replace the abelian group G with a sheaf of abelian groups F over a space X to obtain the complex C (N(U), F). The homology groups of this complex are called the cohomology groups of the open cover U with sheaf coefficients F and are denoted by Ȟq (U, F). Although we would like if Ȟq (U, F) agreed with H q (F), they may not as a short exact sequence of sheaves doesn t necessarily give a long exact cohomology 30