1 Appendix A. Basic homotopy. Let M, N be finite dimensional manifolds. We choose a point x0 M and a point y0 N; they are called the basepoints of M and N. We denote Γ(M,N) the space of all∈ smooth functions f :∈M N. (By → smooth we mean continuous and r-times differentiable, with 0 r ). We denote Γ∗(M,N) the subspace ≤ ≤∞ of Γ(M,N) consisting of functions that preserve basepoints, i.e. f(x0)=y0. We say that two maps f,g Γ(M,N) are homotopic (and write f g) if there exists a continuous map F : M I N such that F (x,∈0) = f(x), F (x, 1) = g(x). Intuitively,≃F gives a one parameter family of maps, depending× → continuously on t, that interpolates between f and g. Sometimes it is convenient to put into evidence the dependence on the parameter, and write ft =F (., t); then f0 =f, f1 =g. In the case when M, N have basepoints and f,g Γ∗(M,N)∗ one requires F (x ,t) = y for all t. If f f are maps from ∈ 0 0 1 ≃ 2 N to P and g1 g2 are map from M to N, then f1 g1 f2 g2. The relation≃ of being homotopic is an equivalence◦ relation.≃ ◦ The quotient of Γ(M,N) by this relation, i.e. the set of homotopy classes of maps from M to N, is denoted [M,N]. Similarly one defines [M,N]∗, the set of homotopy classes of basepoint-preserving maps. The set of homotopy classes thus defined do not depend on r, the class of differentiability of the maps. In fact, from the mathematical point of view, it is most natural to assume that M and N are only topological spaces and that the maps are only continuous (r =0). Two spaces M and N are said to have the same homotopy type if there are maps f : M N and → g : N M such that f g IdM and g f IdN . It is easy to see that if M and N have the same homotopy→ type, then [P,M◦ ] =≃ [P,N] and [◦M,Q≃] = [N,Q] for all spaces P , Q. A space N is said to be contractible if it is homotopy equivalent to a point or in other words if the identity map is homotopic to the constant map. Stated more explicitly, this means that there is a continuous map F : I N N such that × → F (0,y)= y and F (1,y)= y0. For example, all vectorspaces are contractible. It is enough to take the origin as basepoint and consider F (t,y) = ty. If N is contractible, then [M,N]∗ is the trivial set consisting of a single element. To see this it is sufficient to note that for any map f : M N, IdN f = f is homotopic to → ◦ y0 f = y0. So from the point of view of homotopy a contractible space is equivalent to a single point. ◦ M m+1 2 2 In the case when M is a sphere S = x R x1+ + xm+1 =1 with m 1, the sets of homotopy classes can be given a group structure. This{ ∈ case is so| import· · ·ant that it deserves} a≥ special name: the space m πm(N)=[S ,N]∗ is called m-th homotopy group of N. We first show how the group structure is defined in the case m=1 (π1(N) is also called the fundamental group of N). We think of S1 as an open interval I =[0, 1] with the endpoints identified; the basepoint of S1 1 corresponds to 0 (or 1). A basepoint preserving map f : S N is just a loop starting and ending at y0. Given two loops f , f we can define a third loop f f by “going→ first round f , then f at double speed”: 1 2 1 · 2 1 2 1 f1(2t) for0 t 2 f1 f2(t)= 1 ≤ ≤ (A.1) · f2(2t 1) for t 1 . − 2 ≤ ≤ If we denote [f] π (N) the homotopy class of the loop f, then [f ][f ]=[f f ] defines a group multiplication ∈ 1 1 2 1 · 2 in π1(N). In the case m 2, we think of Sm as the m-cube Im with all points of the boundary identified. Note ≥ m m that if we call t1,...,tm the coordinates in I , the boundary ∂I of the cube consists of all points for which m m at least one of the coordinates is equal to 0 or 1. A map f : I N such that for all x ∂I , f(x)=y0 can m → ∈ be regarded as a map f : S N, and thus defines a homotopy class in πm(N). We define f f by → 1 · 2 1 f1(2t1,t2,...,tm) for0 t1 2 f1 f2(t1,...,tm)= 1 ≤ ≤ (A.2) · f2(2t1 1,t2,...,tm) for t1 1 . − 2 ≤ ≤ The group structure in πm(N) is then defined as in the case m=1. The groups πm(N) for m 2 are always abelian, whereas π1(N) need not be abelian. The following sequence of drawings is the proof≥ of this statement in the case m = 2. The first square represents the homotopy between f1 (yellow) and f2 (red), as given in (A.2). Black areas (including the contours of the rectangles) are points where the value of the function is y0. By a continuous sequence of deformations one arrives at interchanging the order of f1 and f2 in the homotopy. It is also immediately clear why this cannot be done for m=1. 2 0 The definition of πm(N) given above works also in the case m =0. The sphere S consists of the two points +1 and 1. One of them, for example +1, can be taken as basepoint. A basepoint-preserving map 0 − f : S N maps +1 to y0 and 1 to some point y of N. Thus there is a bijective correspondence between → − ′ ′ ′ Γ∗(M,N) and N. Two maps f and f are homotopic if y =f( 1) and y =f ( 1) belong to the same arcwise 0 − − connected component of N. Thus π0(N) = [S ,N]∗ = arcwise connected components of N . This set does not have a group structure in general. { } Summarizing, the homotopy groups give some information about the topology of a manifold. π (N)=0 0 6 if N has more than one connected component, π1(N) = 0 if N is multiply connected, πm(N) = 0 if N contains non-contractible m-spheres. One can prove that6 if M is a smooth manifold then the homotopy6 groups characterize its homotopy type. If f : N Q is a smooth map, there are natural homomorphisms πk(f) : πk(N) πk(Q) for all k, → k → defined as follows: πk(f) maps the homotopy class of a map g : S N to the homotopy class of the map f g : Sk Q. One can easily check that these are homomorphisms.→ ◦ → n We will mostly need the homotopy groups of the spheres, πm(S ). These can be conveniently summa- rized in a table: Space n =1 n =2 n =3 n =4 m =1 Z 0 0 0 m =2 0 Z 0 0 m =3 0 Z2 Z 0 All the elements on the diagonal are Z by a theorem of Hopf. All the elements above the diagonal are zero. The elements below the diagonal are generally nonzero and are the tricky ones. The only one that we 3 shall need is π2(S ), which was also worked out by Hopf. It generator is the homotopy class of the Hopf map (the projection of the Hopf bundle). 1 i i ··· 1 n Let M and N have the same dimension n. We denote ω = n! ωi1 indy dy a volume-form on α ∧···∧β N. For example, if N is endowed with a riemannian metric h = hαβdy dy it is natural to consider the ⊗ riemannian volume form ω =√det h dy1 dyn. Given a map ϕ : M N we define the winding number of ϕ ∧···∧ → ∗ M ϕ ω 1 n µ ···µ 1 n W (ϕ)= = d xε 1 n ∂ ϕ ∂ ϕ ω ··· . (A.3) R µ1 µn 1 n N ω Vol(N) ZM · · · R The geometrical meaning of this quantity can be understood as follows. Recall that a point x M is a α∈ regular point for the map ϕ if the tangent map T ϕ x is surjective (i.e., in coordinates, if det(∂µϕ )(x)=0). − 6 A point y Imϕ N is regular value for ϕ if all the points in its pre-image ϕ 1(y) are regular points. It can be proven∈ that if⊂y is any regular value of ϕ, then −1 α −1 α W (ϕ)= # of points in ϕ (y) with det(∂µϕ )>0 # of points in ϕ (y) with det(∂µϕ )<0 . (A.4) − In particular the winding number is the integer topological invariant that characterizes the maps from Sn to Sn. 3 Appendix B. Homology and cohomology Homology and cohomology groups are topics of algebraic topology. It is therefore most appropriate to discuss them in the context of topological spaces. However, since in this book we work only with differentiable manifolds, it is simpler to define these notions in the context of differential geometry. We will define homology using the notion of boundary of a manifold. To define the p-th homology group of a manifold M we consider the set of all p-dimensional oriented submanifolds of M.