Appendix: A Proof of De Rham's Theorem §1. Introduction In Chapter I we mentioned that some of the motivating ideas for the develop­ ment of homology theory in the Nineteenth century arose in connection with such topics as Stokes's theorem, Green's theorem, Gauss's divergence theorem, and the Cauchy integral theorem. De Rham's theorem may be looked on as the modern culmination of this particular line of thought. It relates the homology and cohomology of a differentiable manifold to the exterior differential forms on the manifold. Exterior differential forms are objects which can serve as integrands of line integrals, surface integrals, etc., such as occur in the statement of the classical Green's theorem and Stokes's theorem. De Rham's theorem is of obvious importance, because it is a connecting link between analysis on manifolds and the topological properties of manifolds. In this appendix we will assume that the reader is familiar with the basic properties of differentiable manifolds, differential forms on manifolds, and the integration of differential forms over (differentiable) singular cubes. These topics are explained in many current textbooks, and there would be little point in our repeating such an exposition here. As examples of such texts, we list the following: M. Spivak [6], Flanders [3], Warner [9], and Whitney [10]. The first part of this chapter is devoted to using differentiable singular cubes to define the homology and cohomology groups of a differentiable manifold. We prove that in studying the homology and cohomology groups of such a manifold, it suffices to consider only differentiable singular cubes; the nondifferentiable ones can be ignored. 251 252 Appendix: A Proof of De Rham's Theorem Next, we introduce what may be called the De Rham cacha in complex of a differentiable manifold. This cochain complex consists of the exterior differential forms, with the exterior derivative serving as the coboundary operator. There is a natural homomorphism from this De Rham complex to the cochain complex (with coefficient group R, the real numbers) based on differentiable singular cubes. This homomorphism is defined on any exterior differential form of degree p by integrating that form over differ­ entiable singular p-cubes. The general form of Stokes's theorem is precisely the assertion that this natural homomorphism is a cochain map. De Rham's theorem asserts that this natural cochain map induces an isomorphism on cohomology. The proof we give of De Rham's theorem is modelled on Milnor's proof of the Poincare duality theorem in Chapter IX. The reader who has worked through that proof should have no trouble grasping the structure of our proof of De Rham's theorem. Curtis and Dugundji [11] have also given a 'proof of De Rham's theorem along somewhat similar lines. §2. Differentiable Singular Chains Let M be an n-dimensional differentiable manifold of class COO (we assume the reader is familiar with this concept). In order to define a differentiable singular cube, we must make use of the fact that the standard unit p-cube, IP = {(Xl" .. ,Xp) E RPI0:s; Xi:s; 1, i = 1,2, ... ,p} is a subset of Euclidean space RP. For p > 0, a singular p-cube T:JP -+ M will be called differentiable if there exists an open neighborhood U of IP in RP and an extension T': U -+ M of T such that T' is differentiable (of class COO). We complete this definition by defining any singular O-cube to be differentiable. Remark: If a singular p-cube T:IP -+ M is differentiable, there will, in general, be many different choices for the open neighborhood U and the extension T': U -+ M. We now introduce the following notation: Q~(M) = subgroup of Qp(M) generated by the differentiable singular p-cubes, D~(M) = Dp(M) n Q~(M), C~(M) = Q~(M)/D~(M). The superscript S in the above notation is intended to suggest the word "smooth." We will refer to C~(M) as the group of differentiable or smooth p-chains of M. Note that C~(M) = Q~(M) = Qo(M) = Co(M). §2. Differentiable Singular Chains 253 Next, observe that if T:1P ~ M is a differentiable singular p-cube, then the faces AiT and BiT, 1 ::; i::; p, are all obviously differentiable singular (p - i)-cubes. It follows that op(T) E Q~_1(M). Thus QS(M) = {Q~(M),op} is a subcomplex of Q(M), and CS(M) = {C~(M)} is a subcomplex of C(M). We will also introduce the following notation: for any abelian group G, CS(M;G) = CS(M) ® G, Ct(M;G) = Hom(CS(M),G), H~(M;G) = Hp(CS(M;G», H~(M;G) = HP(Ct(M;G». We can now state the main theorem of this section: Theorem 2.1. Let M be a differentiable manifold. The inclusion map of chain complexes, induces an isomorphism of homology groups, H~(M) ~ HiM). Corollary 2.2. For any abelian group G, we have the following isomorphisms of homology and cohomology groups: H~(M;G) ~ Hp(M;G), H~(M;G) ~ HP(M;G). The corollary follows from the theorem by use of standard techniques (cf. Theorem V.2.3). Before we can prove the theorem, it is necessary to discuss to what extent the methods and results of Chapters II and IlIon homology theory carryover to the homology groups H~(M; G) for any differentiable manifold M. We will now do this in a brief but systematic fashion. (a) Let M1 and M2 be differentiable manifolds, and let f:M 1 ~M2 be a differentiable maps of class Coo. If T: 1P ~ M 1 is a differentiable singular p cube, in M b then fT: 1P ~ M 2 is also differentiable. Hence we get an induced chain map f# :CS(M 1) ~ CS(M2 ) with all the usual properties. (b) Two differentiable maps fO'/1:M 1 ~ M 2 will be called differentiably homotopic if there exists a map f:1 x M1 ~ M2 such that fo(x) = f(O,x) and f1(X) = f(i,x) for any x E M 1, and in addition, there exists an open neighborhood U of 1 x M 1 in R x M 1 and a map f': U ~ M 2 which is an extension of f, and is differentiable of class Coo. The technique of §I1.4 can now be applied verbatim to prove that the induced chain maps fo#, 254 Appendix: A Proof of De Rham's Theorem 11 # : CS(M 1) -+ CS(M 2) are chain homotopic. This has all the usual conse­ quences; in particular, the induced homomorphisms on homology and cohomology groups are the same. (c) An open, convex subset if Rn is differentiably contractible to a point; in fact, the standard formulas for proving that such a subset is contractible are differentiable homotopies in the sense of the preceding definition. From this it follows that if U is an open, convex subset of Rn, then HS(U;G) = {G for p = 0, P ° for p =1= 0, with similar formulas for H~( U; G). (d) Let M be a differentiable manifold, and let A be a subspace of M which is a differentiable submanifold. For example, A could be an arbitrary open subset of M, or A could be a closed submanifold of M. Then we can consider CS(A) as a subcomplex of CS(M); hence we can consider the quotient complex CS(M)/CS(A) = CS(M,A) and we obtain exact homology and coho­ mology sequences for the pair (M,A) using differentiable singular cubes. (e) If T:JR -+ M is a differentiable singular cube, the subdivision of T, Sdn(T) as defined in §II.7, is readily seen to be a linear combination of differentiable singular cubes. Hence the subdivision operator defines a chain map just as in §II.7. Unfortunately, the chain homotopy CPn:Cn(M) -+ Cn+ 1(M) defined in§II.7 does not map C~(M) into C~+ l(M). This is because thefunction 111:[2 -+ [t,1] is not differentiable (the function 110:[2 -+ [is differentiable). However, it is not difficult to get around this obstacle. Consider the real­ valued function 11'1 defined by , ( ) _ 1 + Xl - XIX2 111 Xl,X2 - 2 . - X2 It is readily verified that 11'1 maps [2 into the interval [t,1], and that 111 and 11'1 are equal along the boundary of the square [2. bbviously, 11'1 is differentiable in a neighborhood of [2. Thus if we substitute 11~ for 111 in the formula for Ge(T) in §II.7, then GiT) will be a linear combination of differentiable singular cubes whenever T is a differentiable singular cube. Moreover, the operator Ge will continue to satisfy identities (f.1) to (fA) of §II.7. Thus we can define a chain homotopy IPn: C~(M) -+ C~+ l(M) using the modified definition of Ge . From this point on, everything proceeds exactly as in §II.7. The net result is that we can prove an analog of Theorem II.6.3 for singular homology based on differentiable singular cubes, and the excision property (Theorem II.6.2) holds for this kind of homology theory. §2. Differentiable Singular Chains 255 (f) Suppose that the differentiable manifold M is the union of two open subsets, M= Uu V. Then we can obtain an exact Mayer-Vietoris sequence for this situation by the method described in §III.5. (g) Finally, we note that an analog of Proposition III.6.1 must hold for homology groups based on differentiable singular cubes; this is practically obvious.