An Introduction to the Theory of Obstructions: Notes from the Obstruction Theory Seminar at West Virginia University. Adam C.

An Introduction to the Theory of Obstructions: Notes from the Obstruction Theory Seminar at West Virginia University Adam C. Fletcher c Spring 2009

Contents 1 The Preliminaries: Definitions, Terminology, and Examples 1 1.1 Cells: Building Blocks of Homotopy Theory............ 1 1.2 Building with Cells: Intuition on Complexes............ 2 1.3 Formal Construction of CW-Complexes.............. 6 1.4 Deconstruction of CW-Complexes.................. 7 1.5 Basic Properties of CW-Complexes................. 8 1.6 Deconstructing Further: (Sub)Polyhedra.............. 10 1.7 Deformations and Homotopy.................... 12 1.8 The Homotopy Extension Property................. 14 1.9 Cellular Maps: A New Type of Morphism............. 14 2 Theories of Homology and Cohomology 18 2.1 Terminology of Homology and Cohomology............ 18 2.2 General Homology Theory: Points................. 21 2.3 General Cohomology Theory: Points................ 22 2.4 General Homology Theory: Homotopy............... 23 2.5 General Cohomology Theory: Homotopy.............. 24 2.6 General Homology Theory: Spheres................ 24 2.7 General Cohomology Theory: Spheres............... 25 2.8 General (Co)Homology Theories: Wedges............. 26 2.9 Popular Homology Theories..................... 26 2.10 Popular Cohomology Theories.................... 28 3 Extensibility and Obstruction 30 3.1 Prelude to Obstruction: The Players................ 30 3.2 Prelude to Obstruction: Extensibility................ 31 3.3 Algebraic Tools for Obstructions.................. 32 3.4 Terminology of Homotopy Groups................. 33 3.5 Connectivity and Hurewicz Theorem................ 35 3.6 The Definition of Obstruction.................... 35 1

Abstract The generalization of everyday concepts, and applying rigor thereunto, has long intrigued the mathematician. The topic of an obstruction theory arises from the question of deforming continuous maps, and what, exactly, holes are. Intuitively, the fundamental difference between an orange and a doughnut is that a doughnut has a hole, and an orange does not. When shrinking a rubber band around an orange, the rubber band will vanish; whereas the rubber band that passes through the hole of the doughnut, or the one that encircles the doughnut at its widest point, cannot vanish without cutting the doughnut or snapping the rubber band. When we apply the mathematical rigor that transforms oranges, doughnuts, and rubber bands into spheres, tori, and copies of S 1, we raise the question of what the empty space in the torus actually represents. The shrinking of the rubber band represents continuous deformations of circles and spheres. What, then, will obstruct our continuous deformation? The answer to this question forms the foundation of an obstruction theory.

Chapter 1 The Preliminaries: Definitions, Terminology, and Examples 1.1 Cells: Building Blocks of Homotopy Theory The theory of obstructions should comfortably couched in the arms of homotopy theory, since we wish to discuss topological spaces, the maps of continuous functions, and how these maps are attached to the spaces. In order to study the loops, discs, and spheres that we wish to, we must define some basic homotopic ideas. The first such idea is that of cells. Definition 1.1.1. An open n-cell is a subspace, e n, of a topological space X such that e n is homeomorphic to an open n-sphere, the patch of Euclidean space of all points less than one unit from the origin. It bears mentioning, here, that cells are not the only building blocks of topological spaces. Depending on the branch and purposeful application of the mathematics involved, one could consider blades, which represent n-dimensional triangles (points, arcs, triangles, tetrahedra, etc.), instead. Mathematical physicists often make use of cubes, which are defined by inductively connecting 2n of the n-dimensional cubes at right angles to each other (points, arcs, squares, cubes, hypercubes, etc.). In a topological since, however, these bases are equivalent. It is only when considering the intrinsic geometry at n 2 that we see a difference between the cell, the blade, and the cube; the n-sphere, the n-triangle, and the n-square. 1

CELLS BLADES CUBES points in space: 0!cells open arcs in space: 1!cell boundary is the 0!sphere open discs in space: 2!cells boundary is the 1!sphere open spheres in space: 3!cells boundary is the 2!sphere Figure 1.1: Low Dimensional Building Blocks 1.2 Building with Cells: Intuition on Complexes Now that we have the building blocks of a homotopy theory, we must mix our mortar and build our homotopic foundation. Let us begin by doing so intuitively, and then applying some rigor. Definition 1.2.1. Let X 0 be called the 0-skeleton of a topological space, X, and consist of a collection of 0-cells of X. Definition 1.2.2. The n-skeleton, X n, of a space X is built inductively by attaching a number of n-cells, e n α, (where α is an element of an indexing set) to X n 1 by objects called attaching maps, denoted ϕ n α : S n 1 X n 1, which identify the boundaries of each n-cell to the previous skeleton in some prescribed manner. One can then create a finite complex, X n, or an infinite complex X = n N X n. Although spaces of this latter type form an infinite class, this mega-union seems to be the only such class of infinite complexes. Since this is the case, and the class of finite complexes is fair richer, it is with the finite case that we concern ourselves. Example. Consider, as an example, the construction of the two-sphere, S 2, as the union of a 0-cell and a 2-cell. The attaching map, ϕ 2, attaches the boundary of e 2 (which is S 1 ) to the single 0-cell, as shown in Figure 1.2. 2

2!cell 0!cell A circle is the boundary of a disc Figure 1.2: Example: Cell Structure of the Sphere, S 2 Figure 1.3: Example: Cell Structure of the Torus, T 2 3

Example. Consider the slightly more complicated construction of the twotorus, T 2. To our end, we consider the so-called flat-torus, which is represented as the unit square, I 2, under identifications. We think of the interior of the square as a 2-cell (a disc). We then form the torus by identifying the top to the bottom of the square, and the left side to the right side of the square. That is, ϕ 1 1, shown in red in Figure 1.3, maps one identified 1-sphere to the meridian, and ϕ 1 2, shown in blue in the same figure, maps the other identified 1-sphere to the equator. Finally, the identification of the edges induces the identification of all four corners of the square as a single point. Thus, ϕ 0 maps the identified 0-sphere to the point of intersection on the torus of the 1-spheres. Now, for each n-cell, the attaching maps connect the boundary of the cell to a lower skeleton. The n-cell itself, however, must also be attached to the space in question. This map is called the characteristic map, is denoted Φ n α : e n α X (where the overline denotes topological closure), and extends the attaching map, ϕ n α, by mapping the disc homeomorphically onto the interior of e n α. Example. Returning to Figure 1.2 above, we see that the characteristic map of the sphere is given by Φ 2 : e 2 S 2 that contracts the boundary of e 2 to e 0 via a straight-line homotopy. Example. The torus in Figure 1.3 grants Φ 2 : e 2 T 2 defined by the identification of edges and, subsequently, Φ 1 1 and Φ 1 2 : e 1 i S 1 for i = 1, 2, by identifying endpoints of each edge. Both of the these examples, however, have something in common. Both are orientable, in that the sphere and the torus both have an outward-pointing tangent vector field. In topology, we need to consider as many different types of spaces as possible, and so, we turn to the non-orientable surfaces. One such example is the projective plane. The two-dimensional real projective plane has been equivalently described as the result of identifying all four edges of a square to a single edge, with opposite pairs attached with opposite orientations; as a bi-gon (a circle with two pinched points) with opposite edges identified with opposite orientations; as identifying all lines through the origin in R 2 to a single line; or identifying antipodal points of S 1 to each other. We can take these ideas, most preferably the last, and 4

0 0 S and RP SPHERES PROJECTIVE PLANES 1 1 S and RP 2 2 S and RP Figure 1.4: Constructing Real Projective Space generalize to all n N, resulting in an n-dimensional real projective space, the first three of which are shown in Figure 1.4. For our purposes, however, we wish to see these real projective spaces as having a complex structure. Accordingly, we build the projective spaces inductively. First, RP 0 is merely the identified point, e 0. As a union of the identified edge and its base point, RP 1 is e 1 e 0. Similarly, RP 2 consists of the identified hemisphere and its equator, which is a copy of RP 1. Thus, RP 2 = e 2 e 1 e 0. This pattern, then, generalizes to all n, where RP n is the identified hemi-(n 1)-sphere and its equator, resulting in n RP n = e i. Example. As for the characteristic maps for the projective plane, we have i=0 Φ 2 : e 2 RP 2 and Φ 1 : e 1 RP 1 given by identifying antipodal points of the respectively dimensioned spheres. 5

1.3 Formal Construction of CW-Complexes Let us now apply mathematical rigor to our idea of a CW-complex. Definition 1.3.1. Recall that a space holds the T 2 separation axiom, or is Hausdorff, if, for any two elements of the space, there are mutually disjoint neighborhoods of these points. That is to say, for each x, y X, there exists a neighborhood U of x and a neighborhood V of y so that U V =. Example. The usual topology in R n is Hausdorff, as is the subspace topology on any X R n. One must be careful, however, not to assume that all spaces are Hausdorff. The finite complement topology (where a set U is open if and only if X U is a finite set) is very rarely Hausdorff. We will require a CW-complex to be Hausdorff, since we wish to regard the space as an embedding in R n. This is the case, since we are building the space from cells, which are (fairly trivial) embeddings into R n. As stated previously, we wish to ignore the case of the infinite cell skeleton. To this end, we will make our second requirement for a space to be a CW-complex that the characteristic maps attaching each closed cell, e α n, to X must map the cell boundaries to a finite number of spheres. This is to say that the image of each attaching map is a finite union of spheres. J.H.C. Whitehead called this property closure-finiteness, as it implies an enclosure of each skeleton within a compact space (compact, both in the literal and the mathematical sense). Example. It is not difficult to see that the examples shown in Figure 1.2 and Figure 1.3 are closure-finite. Example. The following example, due to [10] fails in this closure-finiteness criterion. Consider the space X produced by bounding a two-cell by three onecells, then by adjoining a zero-cell at intervals of 1/n along one of the one-cells, as shown here: Figure 1.5: Munkres Non-CW-Complex Notice that the attaching map of the 2-cell consists of infinitely many 0-spheres, and so the space fails to be closure-finite. The boundary of the 2-cell (that is, the space formed by the union of the 1-cells and 0-cells), however, is closurefinite, since the attaching map of each 1-cell consists of at most two 0-cells. 6

Our final criterion for status as a CW-complex will involve a topology placed on the space. Definition 1.3.2. Let X be a topological space, and let A X. We say that A is closed in X if A e α n is closed in the cellular subspace topology for each n. This induced topology is known as the weak topology on X. In conclusion, we say that a space X is a CW-complex, or has a CW-complex structure on it, if the following three criteria hold: X is Hausdorff; X is closure-finite; and X is embued with the weak topology on its cells. As a matter of interest, it is the weak topology on the space, along with the closure-finiteness of the complex that gives the CW-complex its name. Example. Returning to Munkres triangle (Figure 1.5), we notice that the closure-finiteness is the only criterion which the space fails. The space, as a subset of Euclidean two-space in the usual topology, is Hausdorff; and usuallyclosed sets are weak-closed sets, as the weak topology is finer than the usual n topology. That is, for each closed subset A, for each α and each n, the A e α are closed. Example. The boundary of Munkres triangle (that is, the union of the one- and zero-cells), however, is, indeed, a CW-complex, since the Hausdorff and weak topology criteria are inherited from the greater triangle, and we have discussed the closure-finiteness of this space. 1.4 Deconstruction of CW-Complexes As is the case in many branches of mathematics, our primary aim when introducing a topic is to define an object in a category, to show that it is well defined (which will be shown shortly), to give some examples and properties of the object (which also will be given shortly), and, thence, to break the object into smaller pieces. Although out of order, we shall do this last, first. Definition 1.4.1. A subcomplex of a CW complex, X, is a closed subspace, A X, that is a union of cells of X. The pair (X, A) is called a CW pair. Example. Since the RP n are constructed of unions of skeletons, the RP k are subcomplexes for all k < n. Example. If S n is merely e n e 0, as is the usual construction, none of the e i for 0 < i < n is a subcomplex of S n. This seems to make S n simple, in a manner of speaking. We can, however, think of S n as e n N en S Sn 1, where e N and e S represent the northern and southern hemi-n-spheres, respectively. The (n 1)-sphere should be defined in terms of its hemispheres and lower dimensional spheres, as well. 7

Figure 1.6: CW-Decomposition of a Sphere 1.5 Basic Properties of CW-Complexes In preparation for defining obstructions, we wish to build some necessary groundwork on the properties, both topological and homotopic, of CW-complexes. Some will seem to be extremely important, and others, merely as passing interests. We, however, have a purpose for each, which will become apparent in time. Proposition 1.5.1. A compact subspace of a CW-complex, X, is contained in a finite subcomplex of X. Proof. We follow the method of [4]. Let X be a CW-complex, and let C be a compact subspace thereof. We claim, first, that C intersects only finitely many cells. For a contradiction of this claim, we assume C intersects infinitely many cells, and let x i be the points in C e i for each i N. Further, we define the set S of these intersection points. That is, S = {x 1, x 2,... }. As a disjoint union of points, we have that S is closed in X. Notice, trivially, that S {e 0 α} = S is closed in X 0. Assume, then, that S X n 1 is closed in X n 1, and we consider the closed n-cells, e n α. Since S is closed, and attaching maps are continuous, ϕ 1 α (S) is closed in e n α. Since the characteristic map is an extension of the attaching map, Φ 1 α (S) = e m β ϕ 1 α (S), for some m, making this inverse image closed in e n α, as well. We have, then, that in the n-skeleton, S X n is closed. Inductively, then, this is the case for all n. Since X is a CWcomplex (and, hence, a union of these skeletons), S is closed in X. By the same argument, any subset of S is closed, and, hence, S carries the discrete topology. Since S is a compact space (any closed subspace of a compact space is), S must be finite. As this contradicts our assumption, we conclude that S cannot be infinite, so C intersects but finitely many cells. 8

Notice that the cell e α n is contained in a finite subcomplex of X, since, for each n, ϕ α (e α n ) is a finite union of (n 1)-and-smaller spheres, which is compact. Thus, by our proposition, e α n is contained in a finite subcomplex of X. Since C is contained in a finite number of cells, it is contained in a finite number of finite subcomplexes of X, which, in turn, is a finite subcomplex of X. Proposition 1.5.2. The finite product of CW-complexes is a CW-complex. Proof. We, again, follow the method of [4]. For our purpose, it will suffice to show that the product of two CW-complexes is a CW-complex, and, thence, we can proceed inductively to the arbitrarily finite case. Let X and Y be CWcomplexes with characteristic maps Φ X and Φ Y, respectively. We claim that (Φ X Φ Y ) : e n α e m α X Y is the characteristic map for the product space. To this end, we must show that X Y has a complex structure on it. Recall that the product of Hausdorff spaces is Hausdorff. Moreover, if the image of attaching maps in X consists of n spheres and the image of the attaching maps in Y consists of m spheres, then the image of the attaching maps in the product space consists of at most mn spheres, and, so, is finite. Finally, we consider the weak topology on the product space. If A X Y, then A π X (A) π Y (A), where π i are the projections into each factor space. If we put a compactified topology on X Y (think the weak topology), having A compact grants that the projections are also closed and compact. Since π X (A) and π Y (A) are compact, they are contained in a finite subcomplex of their respective factor space, namely n e i and i=1 m e j. j=1 We notice, however, that n m e i e j i=1 j=1 n,m i,j=1 (e i e j ). Hence, A is contained in the union of crossed cells, and the intersection of A with a closed cell will result in a closed set in the subspace topology. Thus, a closed subspace of the product space is closed in the weak topology, giving the product space a CW-structure. 9

1.6 Deconstructing Further: (Sub)Polyhedra Although not all topological spaces are CW-complexes, an arbitrary space X could, feasibly, be homeomorphic to a CW-complex, and could have a CWstructure placed upon it. Such spaces are referred to as polyhedra. The well-behaved polyhedra, those that consist of finitely many cells, are called finite polyhedra. Moreover, a closed subset Y of a polyhedron X that consists of a union of cells is called a subpolyhedron, where the complex structure is referred to as the subcomplex. This is consistent with other branches of mathematics, where a portion of a full object also has the properties of the mother object (cf. subsets, subgroups, submanifolds, etc.). In order to consider continuous deformations of a polyhedron, we will need to consider a homotopy, which can be represented as a continuous function through a time variable. To this end, we consider the space crossed with the time variable, or X I, where I is the unit interval. We will call this space the cylinder space of the polyhedron, X, denote it by Cyl(X), and note that Cyl(X) is, itself, a polyhedron. This is a result of the fact that, if X has a complex structure in terms of the e n α, then the sets e n α 0, e n α 1, and e n α (0, 1) form a complex structure on the cylinder X I, due to Proposition 1.5.2. The union of these crossed cells, then, gives that X I is generated by the cells of X crossed with the cells of I, which are 0, 1, and (0, 1). Figure 1.7: Simplified Cylinder Space 10

Now that we have a complex structure on the cylinder space of a polyhedron, we should study the properties of continuous functions working on this new polyhedron. Doing so will allow us to use the tools of homotopy theory on all polyhedra. We begin with the following named theorem. Proposition 1.6.1 (Borsuk s Lemma). Let Y be a subpolyhedron of a polyhedron X. Then there is a continuous map Γ : X I (X 0) (Y I) that serves as the identity on the range. Figure 1.8: Cylinder Space for Borsuk s Lemma Borsuk s Lemma could effectively be called the Tank Draining Lemma. The essential (and intuitive) idea behind this lemma is that whatever the fish do in the full fish tank will eventually be mirrored on the floor of the empty tank, after it is drained, with the possible exception of the inner storage chamber. As time, t, decreases from one to zero, X Y is continuously deformed to the initial state (X 0), whereas Y remains unchanged. Proof. Let X be a polyhedron, and Y be a subpolyhedron of X. A standard proof technique in the realm of polyhedra is induction on both number and dimensionality of cells. We proceed in this wise, following [1]. We first consider the case when Y = X e n for some n. If n = 0, then the map { (x, 0) if x / Y Γ(x, t) = (x, t) else 11

is the desired map. Let n > 0, then, and consider the attaching map ϕ n : e n = S n 1 X n 1 of the n-cell, and the corresponding characteristic map Φ n : e n X of the disc. We will denote by ι : I I the identity map on the unit interval, [0, 1]; and given by (Φ n ι) : e n I X I (Φ n ι)(, t) = (Φ n ( ), t) is a characteristic map on the cylinder space of the polyhedron. Now, we define ψ : e n I (e n 0) (S n 1 I) by the mapping (, t) (, tχ( )), where χ is a (analytic) characteristic map, taking on the values of 0 and 1 pertaining to whether or not (e n ). We notice that ψ serves as the identity on the range. Let (x, t) X I. Since X is a polyhedron, ((Φ n ) 1 (x), t) e n I. Then and ψ(((φ n ) 1 (x), t)) (e n 0) (S n 1 I) (Φ n ι)(ψ(((φ n ) 1 (x), t))) (X 0) (Y I). We must recall that multiplication and analytic-characteristic functions are continuous. Notice that this function serves as the identity on its range and is continuous, since the inverse of a local homeomorphism, product of continuous functions, and compositions of continuous functions are all continuous. We define Γ to be this function. In the general case, we build a larger Γ by iterating the previous Γ over all 0-cells, then 1-cells, 2-cells, and so on, in X Y. 1.7 Deformations and Homotopy In the effort to apply rigor and dimensionality to the idea of shrinking rubber bands on our spaces, we need to define the continuous deformation of spheres. Definition 1.7.1. Let X be a topological space and Y be a subspace of X. There is a continuous deformation of X onto Y if there is a collection of continuous maps f t : X X for t I so that f 0 = X, f 1 = Y, and f t is the identity when restricted to Y, for all t I. 12

Definition 1.7.2. Let X and Y be a topological spaces. There is a homotopy of X to Y if there is a collection of continuous maps f t : X Y for t I and a continuous map H : X I Y so that H(x, t) = f t (x), f 0 = X, and f 1 = Y. Notice that a continuous deformation is just a homotopy of a space to one of its subspaces. The collection of equivalence classes of these deformations/homotopies on the space X form an algebraic group, called the fundamental group, of the space, which is denoted π 1 (X). It is worthwhile for the reader to examine, both intuitively and algebraically, the deformations represented below. Figure 1.9: Discs are Nullhomotopic Notice that the unit disc can be continuously deformed to its origin by the straight line homotopy, given by h(x, t) = x(1 t). Any topological space that can be continuously deformed to a point is said to be nullhomotopic. Figure 1.10: The Punctured Plane is Homotopic to the Circle Furthermore, notice that the so-called punctured plane, the space R 2 {0}, can be continuously deformed to the unit circle, again, by the straight line homotopy, this time give by h(x, t) = x(1 t) + where is the Euclidean norm on R 2. x x t, 13

The Möbius band, a non-orientable surface, can be continuously deformed to its core circle, as well. Again, this is done by a straight-line homotopy that maps each point of the Möbius band orthogonally onto the core circle. The punctured torus can be continuously deformed to a wedge of two 1-spheres. This can be visualized using the flat torus, as in Figure 1.3. The punctured flat torus can be seen to be very similar, then, to the punctured plane, in Figure 1.10. This retracts the interior of the flat torus to its boundary edges, each of which represent a circle. These circles are joined at the identified point, creating the topological space S 1 S 1. 1.8 The Homotopy Extension Property Borsuk s Lemma gives us a way to extend (or contract) a given continuous map from a larger space to a smaller space in a continuous manner. It is, however, a fairly specific lemma and is reliant on the cylinder space of the contrived specific space. We wish to generalize this extendibility to a broader category of spaces without relying as much on cylinder spaces. The homotopy extension property is, therefore, the foundational property for the study of extensions. Proposition 1.8.1 (Homotopy Extension Property). Let X be a complex, and let Y be a subcomplex, thereof. Furthermore, let Z be a general topological space. Given f 0 : X Z and a continuous deformation g t : Y Z for t I, with f 0 = g 0 on Y, then there is a continuous deformation f t : X Z of f 0 so that f t = g t for all t I on Y. Proof. Let X, Y, Z, f 0, and g t be as defined in the conditional. Denote by ĝ : Y I Z the continuous deformation defined by ĝ(y, t) = g t (y). By Borsuk s Lemma, there is a continuous Γ = ˆf : X I (X 0) (Y I) that is the identity on the range. That is, ˆf(x, t) = ĝ(x, t) whenever x Y or when t = 0. If x Y, then, ˆf and ĝ agree; making ˆf an extension of ĝ. The restriction that ˆf(x, 0) = ĝ(x, 0) gives that f 0 = g 0. Thus, the extension preserves the deformations by g and the function f 0 = g 0, as desired. 1.9 Cellular Maps: A New Type of Morphism As a brief interlude into category theory, we consider our position. We have a collection of objects (our CW-complexes), and we have broken them down (into subcomplexes). It seems that we wish to study continuous mappings on these objects. Better yet, we need to studying continuous mappings between these objects. When working with numerical, set theoretic, or algebraic mathematics, we often choose operations (plus, minus, times, divide; intersection, union, cartesian product; tensor products); when working with geometry or topology, 14

we choose operators (divergence, gradient, curl; differential forms). In the category of cell complexes, our morphism will need to be a continuous mapping between two objects. Definition 1.9.1. Let X and Y be two cell complexes. A cellular map is a continuous function f : X Y so that the image of each skeleton in the domain is contained within the corresponding dimensional skeleton of the range. That is to say, f(x n ) Y n for all n. Cellular maps serve two larger purposes: first, after statement of the next proposition, we can approximate any topological space by a CW-complex, and, so, we can simplify the way we see some complicated spaces. Additionally, we will study relationships between several different types of continuous maps, most likely on CW-complexes (when discussing higher-dimensional homotopy groups and [co]homology classes). We will consider these as cellular maps. Proposition 1.9.2 (Cellular Approximation Theorem). Let X and Y be cell complexes. A continuous map f : X Y is homotopic to a cellular map. Moreover, if f is cellular on a subcomplex X of X, the homotopy is constant on X. One should note here that in other texts, many authors revert to the cellular approximation theorem after developing homology theory, and so use relative homology to prove a generalized (relative) case of the Cellular Approximation Theorem first, with this version as a corollary. We follow [1], however, and prove the theorem directly. Proof. As has been seen before, we will proceed by induction on cells, both by order and by number. We notice that if X and Y have only zero-skeletons; that is to say, that X and Y are discrete point-sets, then the proof is trivial. Since the image of any map between the two will fall within the zero-skeleton of Y, it is a cellular map. Consider the inductive case, where X is the result of an n-cell being joined to a subcomplex, X, and Y is the result when an m-cell (where m > n) is joined to a subcomplex, Y. Using the characteristic maps, we chart a system of coordinates onto e n and e m and define a function f 0 : X Y. For those readers familiar with the study of manifolds, this will be similar (but not congruent!) to sending charts from R n into the manifold [9, 2]. 15

X e n e m Y Figure 1.11: Construction of f 0 for Cellular Approximation The Map: Define f 0 to be the composition of the following maps: (Φ n ) 1 : X e n, the characteristic map for the selected n-cell; ι : e n e m, the inclusion map; and Φ m : e m Y, the characteristic map for the selected m-cell; with f 0 (X ) Y (for the induction process). By construction, then, f 0 is a composition of continuous functions and is, therefore, continuous. We may, then, approximate f 0 by differentiable functions (and, in particular, polynomials) via the Weierstrass Approximation Theorem. The Approximation and Deformation: We denote by f : X Y this Weierstrass approximation of f 0. Since the inclusion ι : e n e m is proper, due to strict inequality of dimension, there is a y 0 Y that is not in the image, f (X). Let x 0 (Φ m ) 1 (y 0 ) e m, and let h t be a deformation of e m {x 0 } onto itself that is constant on the boundary of the closed cell; that is, S m 1. The Homotopy: For t [0, 1], let { f t (Φ m h t (Φ m ) 1 f )(x) for f (x) Φ m (e m ) =. f (x) otherwise Notice that f 0 = f (since h 0 is an identity) and that f t gives a continuous deformation of f to f 1, both of which map X Y. Notice, moreover, that f 1(X) Y, and, by construction, is constant on X. Our proposition, then, is proven in the special case; and induction on cells concludes the proof. 16

As another note of interest, a map f is said to be regular at a point x if its coordinate maps (built from the characteristic maps as above) have continuous first-order partial derivatives. This is equivalent to the nonsingularity of the Jacobian matrix. If the Jacobian determinant is positive or negative, so, too, is the point denoted. It can be shown (in a manner similar to the proof of the Cellular Approximation Theorem), that cellular maps can be approximated by regular maps. 17

Chapter 2 Theories of Homology and Cohomology 2.1 Terminology of Homology and Cohomology Definition 2.1.1. A topological space X on which a CW-complex structure can be placed is called a polyhedron. A subspace of X on which a CW-complex structure can also be placed is called a subpolyhedron. Definition 2.1.2. Let G be a set of objects, and let g, h, k G. Let be the product operation on G. The set G is said to be closed under if g h G for all g, h. The set G is said to be associative under if g (h k) = (g h) k for all g, h, k. An identity of the set is an element e G so that e g = g e = g for all g. An inverse of an element g is an element g 1 G so that g g 1 = g 1 g = e. The set G is said to be an algebraic group if the set is closed and associative under its operation, contains an identity, and contains inverses for all its elements. Definition 2.1.3. Let G and H be algebraic groups. A map f : G H is said to be a homomorphism of groups if f(xy) = f(x)f(y) for any x, y G. This is to say, a homomorphism preserves products from group to group. As a matter of notation, unless otherwise defined, we shall, throughout this chapter, let X and Y be polyhedra, and let Z be a subpolyhedron of X. We will denote by H n (X) an algebraic group, called the n-th homology group, and by H n (X) another algebraic group, called the n-th cohomology group. The reader should be forwarned that [5] refers to this group as the contrahomology group, instead. We, further, consider general homomorphisms f : X Y ; and their induced homology (respectively, cohomology) homomorphisms f : H n (X) H n (Y ) and f : H n (Y ) H n (X). Between these groups, we also have the boundary and coboundary maps : H n (X/Z) H n 1 (Z) and δ : H n (Z) H n+1 (X/Z). 18

Now, in order for us to consider homology and cohomology groups, we should have a given theory within which we may work. The framework for the general homology or cohomology theory is given by a list of axioms, or postulates, that will allow us to base our proofs on facts. One must be careful in interpretation, however. The following definitions include the axiom system of a general homology (or cohomology) theory. In order to show that a certain homology theory is valid, one must show that the listed axioms do, indeed, hold. This being said, let us define our theories. Definition 2.1.4. Using the notation above, a collection of polyhedra, groups, homomorphisms, and boundary maps form a homology theory if and only if the following criteria are met: Composition of induced homomorphisms is well-behaved: (g f) = g f Identities are preserved by induction: id : H n (X) H n (X) is the identity automorphism Diagrams commute; that is, f = g, for the homomorphisms below X with subspace Z f Y with subspace T g H n(x/z) f * H n(y/t) boundary H n!1(z) g! * H n!1(t) boundary Figure 2.1: Diagrams Commute in Homology Theories Homotopy is perserved by induction: f g f = g Homology pair sequences are exact H n (S 0 ) = 0 for all n 0. The nontrivial homology group, called the coefficient group, will be denoted G. 19

Definition 2.1.5. In a similar manner, we define a cohomology theory by the following criteria: Composition of induced homomorphisms is devaheb-llew : (g f) = f g ; that is to say, well-behaved in the contravariant sense Identities are preserved by induction: id : H n (X) H n (X) is the identity automorphism Diagrams commute; that is, f δ = δ g, for the homomorphisms below X with subspace Z f Y with subspace T g H n(x/z) f* H n(y/t) boundary n!1 H (Z) g* n!1 H (T) boundary Figure 2.2: Diagrams Commute in Cohomology Theories Homotopy is perserved by induction: f g f = g Cohomology pair sequences are exact H n (S 0 ) = 0 for all n 0. The nontrivial cohomology group, called the coefficient group, will be denoted G. In the following sections, we will prove certain computational results that hold for all homology (respectively, cohomology) theories. We should recall that not all collections of spaces, morphisms, and boundary operators will form a homology or cohomology theory. The following, however, will hold whatever the fixed theory is. 20

2.2 General Homology Theory: Points In a general homology theory, the following results regarding topological points and sets of points are true. Proposition 2.2.1. Let X = {x} be a one-point space. Then, for each n, we have H n (X) = 0. Proof. Consider {x} {x}, the map i : {x} {x}, and the quotient map j : {x} {x}/{x} = {x}. Then the exact homology sequence yields H n ({x}) i H n ({x}) j H n ({x}/{x}) = H n ({x}) Since i and j are identity automorphisms, they are isomorphisms. In particular, j is injective, and its kernel is 0. Since the sequence is exact; that is to say, since im(i ) = ker(j ), we have that the image of i is also 0. As an isomorphism, though, i is surjective; that is, H n ({x}) = 0. Proposition 2.2.2. Let X = {x 1, x 2,..., x k } be a k-point space, with coefficient group G. Then and H n (X) = 0 for n > 0. H 0 (X) = Proof. We proceed inductively. We notice that the statement is true when k = 1, by Proposition 2.2.1. Consider the case, then, when k > 1. Let k 1 i=1 X = {x 1,..., x k } and Y = {x 1,..., x k 1 }. Then X/Y = {x 1, x k }. Let i be the inclusion map, Y X and j be the quotient map, X X/Y. Then, for all n 0, the exact sequence gives to be, inductively, which makes H n (X) = 0, as well. H n (Y ) i H n (X) j H n (X/Y ) On the other hand, if n = 0, we see to be, through induction, G 0 i H n (X) j 0, H 0 (Y ) i H 0 (X) j H 0 (X/Y ) 0 0 k 2 i=1 G i H 0 (X) j G 0. 21

As a split exact sequence, this gives H 0 (X) to be the product of the other two groups; that is, as desired. H 0 (X) = k 1 2.3 General Cohomology Theory: Points In a general cohomology theory, the following results regarding topological points and sets of points are true. Proposition 2.3.1. Let X = {x} be a one-point space. Then, for each n, we have H n ({x}) = 0. Proof. Consider {x} {x}, the inclusion map i : {x} {x}, and the quotient map j : {x} {x}/{x} = {x}. Then the exact cohomology sequence yields δ H n ({x}) i H n ({x}) i=1 G, j H n ({x}/{x}) = H n ({x}) δ Since i and j are identity automorphisms, they are isomorphisms. In particular, i is injective, and its kernel is 0. Since the sequence is exact; that is to say, since im(j ) = ker(i ), the image of j is also 0. As an isomorphism, though, j is surjective; that is, H n ({x}) = 0. Proposition 2.3.2. Let X = {x 1, x 2,..., x k } be a k-point set, with coefficient group G. Then and H n (X) = 0 for n > 0. H 0 (X) = Proof. We proceed inductively. We notice that the statement is true when k = 1, by the result in Proposition 2.3.1. Consider the case, then, when k > 1. Let k 1 i=1 X = {x 1,..., x k } and Y = {x 1,..., x k 1 }. Then X/Y = {x 1, x k }. Let i : Y X be the inclusion map, and j : X X/Y be the quotient map. Then, for all n 0, the exact sequence gives to be, inductively, δ H n (X/Y ) j H n (X) i H n (Y ) δ G δ 0 j H n (X) i 0 δ, which, as a series of isomorphisms, makes H n (X) = 0, as well. 22

On the other hand, if n = 0, we see to be, through induction, δ H 0 (X/Y ) j H 0 (X) i H 0 (Y ) δ 0 0 δ G j H 0 (X) i k 2 i=1 G δ 0. As a split exact sequence, though, this gives H 0 (X) to be the product of the two factor groups; that is, as desired. H 0 (X) = k 1 2.4 General Homology Theory: Homotopy Our study of polyhedra will give us a larger scale on which to study homology theory. We wish, however, to calculate as few homology groups as possible. It is to this end that we appeal to homotopic spaces. Since topology is a study of spaces that can be bent and stretched continuously, we hope that homotopic spaces will have the same homologous qualities. Proposition 2.4.1. Homotopic polyhedra have isomorphic homology groups. Proof. Let X and Y be homotopy equivalent polyhedra. Then, by the definition of homotopy equivalence, there exists a homotopy f : X Y and a homotopy g : Y X so that g f : X X and f g : Y Y are the identities on their respective polyhedra. By the second and first axioms (respectively) of homology theory, we see that and that i=1 G, id Hn(X) = (id X ) = (g f) = g f id Hn(Y ) = (id Y ) = (f g) = f g. Hence, there are inverse automorphisms passing between H n (X) and H n (Y ), making these groups isomorphic. Example. Closed balls, e n, of any dimension, n, have trivial homology groups. Proof. Since there is a deformation retract of e n to a point (the straight line homotopy), the closed ball and a point have the same homology groups. Since, in Section 2.2, we found that the homology groups of a point are always trivial; hence, so, too, are those of a closed ball. 23

2.5 General Cohomology Theory: Homotopy In a similar manner to the previous section, we wish to show that cohomology is also preserved by homotopies. The proof is identical, mutatis mutandi. Proposition 2.5.1. Homotopic polyhedra have isomorphic cohomology groups. Proof. Let X and Y be homotopy equivalent polyhedra. Then, by the definition of homotopy equivalence, there exists a homotopy f : X Y and a homotopy g : Y X so that g f : X X and f g : Y Y are the identities on their respective polyhedra. By the second and first axioms (respectively) of cohomology theory, we see that and that id H n (X) = (id X ) = (g f) = f g id H n (Y ) = (id Y ) = (f g) = g f. Hence, there are inverse automorphisms passing between H n (X) and H n (Y ), making these groups isomorphic. Example. Closed balls, e n, of any dimension, n, have trivial cohomology groups. Proof. Since there is a deformation retract of e n to a point (the straight line homotopy), the closed ball and a point have the same cohomology groups. Since, in Section 2.3, we found that the cohomology groups of a point are always trivial; hence, so, too, are those of a closed ball. 2.6 General Homology Theory: Spheres As mentioned above, if we are to study the homology of polyhedra, we must study the homology of our cells. Since the Cellular Approximation Theorem allows us to approximate almost any nice topological space by CW-complexes, almost any nice topological space has homology generated by its cell structure. We have now been given the homology of 0 and 1 cells and their topological closures. In order to determine the homology of other complexes, we must study the boundaries of these closed cells, the spheres. Proposition 2.6.1. Spheres, S n, of any dimension, n, have trivial homology groups for all k n, and have homology group G for k = n, where G = H 0 (S 0 ). Proof. Notice that, by our homology axioms, the statement is trivial (assininely so) for n = 0. For our sequence, consider X = e n and its boundary, Y = S n 1. Then the quotient space, X/Y = S n, by the standard construction of spheres. Again, we consider i : Y X as the inclusion map, and j : X X/Y as the quotient 24

map. This gives our homology exact sequence (often called the Mayer-Vietoris sequence), then, to be H k (Y ) i H k (X) j H k (X/Y ) H k 1 (Y ) i H k 1 (X) j H k 1 (X/Y ), which becomes, by substitution, H k (S n 1 ) i H k (e n ) j H k (S n ) H k 1 (S n 1 ) i H k 1 (e n ) j H k 1 (S n ). By the homology of points (and all things homotopic thereunto), we have that H k (S n 1 ) i 0 j H k (S n ) H k 1 (S n 1 ) i 0 j H k 1 (S n ). is exact. The exactness of the sequence grants that H k (S n ) = H k 1 (S n 1 ) for each n, k. Moreover, repeated iterations of this isomorphism grant that H k (S n ) = H k n (S 0 ). We recall, however, that H m (S 0 ) is trivial for all m 0 (which is when k n), and is G when m = 0 (or k = n), as desired. 2.7 General Cohomology Theory: Spheres For the same reason as above regarding the homological nature of spheres, we desire to extract the cohomological nature of spheres. The result, not surprisingly, is the same. Proposition 2.7.1. Spheres, S n, of any dimension, n, have trivial cohomology groups for all k n. Moreover, these spheres have cohomology group G for k = n, where G = H 0 (S 0 ). Proof. Again, by the cohomology axioms, we roll our eyes at the n = 0 case. For our sequence, consider X = e n and its boundary, Y = S n 1. Then the quotient space, X/Y = S n, by the standard construction of spheres. As per usual, we consider the inclusion map, i, and the quotient map, j. This gives our cohomology exact sequence (often called either the Mayer-Vietoris or, misleadingly, the de Rham sequence), to be δ H k 1 (X/Y ) j H k 1 (X) i H k 1 (Y ) δ H k (X/Y ) j H k (X) i H k (Y ) δ, which becomes δ H k 1 (S n ) j H k 1 (e n ) i H k 1 (S n 1 ) δ H k (S n ) j H k (e n ) i H k (S n 1 ) δ. By the above results, the cohomology of a closed ball is the same as that of a point, so the sequence becomes δ H k 1 (S n ) j 0 i H k 1 (S n 1 ) δ H k (S n ) j 0 i H k (S n 1 ) δ. The exactness of the sequence grants that H k 1 (S n 1 ) = H k (S n ) for each n, k. Repeated iterations of this isomorphism grants that H k (S n ) = H k n (S 0 ). We recall that H m (S 0 ) is trivial for all m 0 (which is when k n), and is G when m = 0 (or k = n), as desired. 25

2.8 General (Co)Homology Theories: Wedges Since we wish to study polyhedra, in general, and not merely their cells, piecemeal, we need to somehow combine the cells and consider what happens to their homology or cohomology as a result of said combination. The most common combination is the topological wedge. Definition 2.8.1. Let X and Y be topological spaces. Then the wedge sum of the two spaces, denoted X Y is the so-called one-point union, the space resulting from identifying one point, x 0, of X to one point, y 0, of Y. One should be aware, however, that, although the wedge sum is sometimes called the wedge product, this is not the same as the exterior product,. Although this exterior (wedge) product is important in the study of manifolds and differential forms, it is outside the scope of this chapter. Proposition 2.8.2. If X = p i=1 Sn, then H n (X) = p G i=1 and X has trivial homology groups otherwise. Proposition 2.8.3. If X = p i=1 Sn, then H n (X) = p G i=1 and X has trivial cohomology groups otherwise. Proof. The proof of each of these is the same, mutatis mutandi, and is similar to that of Proposition 2.2.1, where we merely change {x i } to S n i. 2.9 Popular Homology Theories Now that we have the general definition of a homology theory, and have seen some properties held by all such considerations, we define, as a matter of interest, several of the more popular homology theories. The reader should keep in mind, however, that these are not the only homology theories in existence; however, these are the theories that receive the most attention from the mathematical community at large. Homology Theory (Simplicial Homology). A homology theory that considers triangulations of a topological space and the cycles and boundaries of said triangulations. 26

As stated in Chapter 1, a simplex is an n-dimensional triangle. Thus, a simplicial complex consists of an n-triangulation of the space. With each k-simplex, σ i, we can define a k-chain, consisting of formal sums, N a i σ i. i=1 If the k-chains form a loop of sorts within the space; that is to say, if the k-chain has no boundary, and is, therefore, in the kernel of the boundary map, the chain is said to be a cycle. On the other hand, if the chain is the image of some boundary map, we call it a boundary. Homology groups, then, are defined in terms of cycles modulo boundaries. Homology Theory (Singular Homology). A homology theory built on the result that all manifolds of dimension three or lower have a triangulation. Continuous maps send a simplicial structure to a manifold. Such maps form chain complexes, with cycles and boundaries relating thereunto. Singular homology is a homology theory for those topologists who really want to be algebraists. The theory is more categorical and algebraic. Built from simplicial homology, this theory depends on the uncountably many triangulations that map from a simplex to a manifold. Each complex of chains is acted upon by chain maps (induced between chain groups by homomorphisms between manifolds) and boundary operators. In a similar manner to the simplicial homology, the kernel of the boundary operator is defined to be the collection of cycles, and the image of the boundary operator is the collection of boundaries. Again, homology groups are defined in terms of cycles modulo boundaries. Homology Theory (Cellular Homology). A homology theory that considers the attaching of cells to a topological space, the cellular map chains created thereby, and kernels and images of the resultant boundary operators. In a manner similar to that of singular homology, we can consider the mapping of cells to topological surfaces via attaching and characteristic maps. As we have seen above, cellular maps are the desired homomorphisms to accomplish this. Since we wish to define homology classes to be invariant under homotopy, we consider the classes of cellular maps. This, again, results in a chain complex. Induced cellular maps create chains, where boundary operators move from one dimension to the next lowest dimension. Again, kernels and images of these boundary operators are used to define the homology classes of a space. Because of the natural use of spheres and discs in our topological thinking, in the sequel, we shall prefer to use cellular homology. Low-dimensional topologists often use simplicial or singular homologies, as they can be concerned with certain triangulations of a space. (The Euler characteristic and Gauss-Bonnett formulas in geometry are results of a triangulation.) 27

Although we shall not prove it here, these three homology theories are equivalent. We refer the reader to [4] for the proof, but point out that the equivalence of simplicial and singular homology is dependent on the fact that both require triangulation. Since homology is preserved by homotopy (as in the axioms), and, thus, by continuous maps, the homology groups in simplicial homology should be preserved by those of singular homology. Similarly, there is an equivalence between cellular and singular homologies at the categorical level. Both are concerned with the chain map complexes, and these maps are often homotopic. This results in a correspondence between the chain complexes, and, so, the homology groups that are defined by both systems as cycles modulo boundaries. Homology Theory (Morse Homology). A homology theory defined on a smooth (differentiable) manifold with a Riemannian metric, dependent on the critical points of the manifold. This flavor of homology is often used in the study of flows over a smooth manifold, as in thermodynamics, or the like. First of all, for more information than is given here, we refer the reader to [7]. Now, if X happens to be a Riemannian manifold, then it has a tangent space centered at each x X. Thus, the differential, df x, of a smooth function f : X R at x is defined. We say that x is a critical point of f if df x = 0. Moreover, the Morse index of a point is the number of negative eigenvalues of the Hessian matrix (the matrix with second partial derivatives of the function). A chain complex (or module) for Morse homology is built of free Z-modules (or free abelian groups) generated by the index n critical points of our given function. The boundary operator between consecutively indexed chains provides a count of the gradient flow lines. For a proof that the Morse homology is, indeed, a valid homology theory, as well as for a number of examples of its use, we refer the reader to [7]. It is a rather amazing fact, which makes up the body of the third chapter of [7], that Morse homology is, in fact, isomorophic to singular homology; and, therefore, to the other two previously mentioned homologies. Homology Theory (Floer Homology). A generalization of Morse homology that uses tools of gauge theory and symplectic geometry. This is the flavor of the month in terms of current research in homology theory, and where, in eventuality, the author will head. 2.10 Popular Cohomology Theories Similarly, we wish to discuss, as a matter of interest, several of the more popular flavors of cohomology theory. To this end, there are three large categories of cohomology theories. The first of these are the Eilenberg-Steenrod cohomology theories. We shall explore these below. There are also the cohomology theories that loosen one of the Eilenberg-Steenrod axioms, and depend on K-theory and 28