i I DIFFEREN~TALam ALGEBRRIC TOPOLOGY Chapter I: Differentiable Manifolds Seetzoh 1: Topology of Euclidean Space Topological spaces, Continuous functions, Metric spaces, Convergence, Compactness, Heine-Bore1 Theorem, Maxima and minima Section 2: Differential Structure of Euclidean Space Definition of differentiable function, cr-functions, Criterion in terms of partial derivatives, Chain rule, Diffeomorphism Section 3: Manifolds in Euclidean Space Statement of the Implicit-FunctionTheorem, Consequences, Definition of manifold in Euclidean space, Examples, Reformulation of the Implicit Function Theorem, Inverse Function 'Theorem, Proof -Ekt?tXoh\4: Manifolds -- The Abstract Definition Charts, Overlap functions, Atlases, Counter- examples, Definition of Manifold, Dimension Section 5: Examples of Differentiable Manifolds Quotients of free group actions ,T~,wn, apn, homogeneous coordinates, homogeneous polynomials, Lie groups Section 6: Further Notes and Generalizations Other kinds of manifolds--complex, foliated, , Manifolds with boundary, Algebraic varieties 'Sect'ioh 7: Maps between Manifolds Definition of a cr-map, rank of the differential, submersion, immersion, and embedding, submanifold Chapter 11: Tangential Structure Tangent plane at a point in terms of defining equations, in terms of curves, normal plane SBct20rX 2: Tangent Planes in General --* Germs of cm-functions at a point, derivations, TMp, A basis for TMp . b (i ' Wctr'on\ 3 : The Tangent Bundle Local trivialization, Transition functions, Vector bundles, Differential of a cW-map, WRitney sump Sections \ 204 : Orfentability For vector spaces, For vector bundles, For manifolds Sect2'o'nn5 : Vector Fields Definition, Integral curves, Ordinary differential equations, Existence, uniqueness, and Cwvariation with parameter of solutions, Local structure of non-zero vector fields, 1-dimensional foliations, Case of compact manifolds cri6: Algebraic Structure of Vector Fields Topology, Real vector space structure, Module structure over the Cw-functions, Derivations of c"-functions, Lie bracket, Lie algebra Chapter\ ITX: Differentials Forms Sect2on 1: Multilinear Algehra Alternathg forms, (v*) , Determinant, Wedge product SectSon 2: Differential Forms - The Definition Definition, Local expre'ssions, Transformation law, Wedge product, Module structure over the C"-functions, Volume forms for surfaces in 1g3 Section 3 : Integration Definition in local coordinates, Proof of independence of the coordinates, Global definition Section 4: Exterior Differentiation Axioms, Proof of existence and uniuueness, Formula in local coordinates Secki$n. 5: Complex Valued Differential Forms Definition, Decomposition of a complex manif old Sectbn' 6: Manifolds with Corners Definition, Structure of the boundary Statement of Stokes' Theorem Section 8: Partitions of Unity Definition, Proof of existence Sectton 9: Proof of Stokes' Theorem S~eOioh$0: Applications of Stokes' Theorem Proof of Cauchy s Integral Forwla The windi'ny number as an integral, Examples of closed forms on the circle and higher dimensional tori. i I I I Section 1 : THE TOPOLOGY OF EUCLIDEAN SPACE We begin these notes with a brief review of point set topology. We shall, in. the sequel, always be concerned with "nice" spaces (metric I spaces). Thus, we shall not deal here at all with pathological examples Cst but shall restrict ourselves, whenever convenient, to metric spaces. 1 w A topological space, (X,.T), is a set, X, together with a collection, r. r. I : T, of subsets of X called open subsets of X. The collection is 1 required to satisfy the following four axioms : I I 1,) X is an open subset of X. I 4' 2 ) Tfie empty set, +, is an open subset of X. 3 An arbLtrary union of open subsets is an open subset of X. 1 7, 4) . # finite intersection of open subsets is an open subset. The main example that we have in mind in this chapter in Euclidean space, mn. Its underlying set consists of all n-tuples, (xl, .- , xn), =A of real numbers. The topology is defined in terms of the usual distance I, * function in mn : We define the open ball of radius r centered about a point p to be : A set U in lRn is defined to be open if and only if for each , p E U there is E > 0 so that BE(p) CU. One checks easily that this collection of subsets satisfies the four axioms. Z* If X is a topological space, and if A c X is a subset, then A inherits a topology from X, The open sets of A are all inter- sections, A n U, where U is an open subset of X. As an example, if we let lRk lRn be the subset of all n-tuples of the form {(x~,-** , xk, 0, - ., 0) 3, then the topology that mk inherits from lRn is identical. to the topology defined abstractly, as above, for lRk If X and Y are topological spaces, then f : X + Y is a continuous function if and only for every open set U GI, the set f-I (u) c X is open. (Recall that f-I (u) = {x E x I f (x) E u 1. ) A homeomorphism from X to Y is a continuous bijection f : X -+ Y whose inverse f-' : Y -t X is also continuous. As we have seen, the topology of Eln is defined in terms of the Pythagorean distance function. Abstracting the basic properties of this distance function leads to the concept of a metric space. Many of the basic properties of IR" are shared by all metric spaces. Definition : Let X be a topological space. A metric is a continuous function d : X x X -t lR+ = {r E lRl r -> 03 such that : 2 d(x,y) = o if andonly if x=y, and e L - A word is necessary about the topology on X x X. It is the so called - product topology, In general, if A and B are topological spaces, then A x B receives a natural topology - the product topology. A set V =A x B is open if and only if for every p E V, there are open sets, UA of A and UB of B, such that p E (UA x UB) c v. It is an easy exercise to show that the topology on lFtn agrees with i the (n-fold) product topology when we consider IR" as IRx xIR (n-times). The metric on IEZn is, of course, the Pythagorean distance. We. also denote d(x,O) by Ilxll. 00 If X is a metric space and {X,},=~ is a sequence of points of X, then we say that x.converges to x, or {xn) + X, if and only if lim d(xn,x) = 0 . (As an exercise, give the definition n+m of convergence in an arbitrary topological space .) Clearly, a sequence can . converge.', to at most one point in a metric space and need not -4 converge to any point at all. - - Lemma 1.1 : -Let X -and Y -be metric spaces, --and let f : X + Y -be -a function. -Then f -is continuous --if and only -if whenever -a sequence {x converges to x n 1 - & X the sequence.{f (xn)} comerqe$ to f (x) in- y. proof : Suppose that there is a sequence x in X which converges to x but that {f (x,) 1 does not converge to f (x). This means that there is an open ball, BE (f (x)) and a sequence of natural numbers % approaching + so that f (x 1 fi BE (f (x) . Hence, fml (BE (f (x)) ) "k contains x but does not contain any xn . Since {x.~14 x. this k implies that no open ball, B6 (x), is contained in fmL(BE(x) . This I shows that f-l (BE (f (x))) is no; open, and consequently that f is not continuous. Conversely, suppose that whenever k,} + x then {f(xn)} + f(x) . Suppose in addition that f is not continuous. From these assumptions we will derive a contradiction. If f is not continuous, then there is an open set U c Y so that f-'(U) c X is not open. Thus, there is x E f-'(U) such that there is no open ball of the form B6 (x) contained in f-l (U) . Thus, for every n > 0, there is a point xn E(X - fw1(U)) such that d(x.,,x) < . The sequence {xn> converges to x. Since xn j! fml (u), we have f (in)fi iJ. Thus, {f (xn)1 does not converge to f(x). This is the sought after contradiction which shows that if €xsn1+ x implies If(x,) 1 + f (x). then f is contir~uous. Examples : 1) Any map f : Wn + W which is given by polynomials in the coordinates (x~,--• , xn) is continuous. Hence the following maps are continuous : n Let d(lRn, *) be the linear maps from lRn to lRm. Any such 2) mapping is identified with an (m x n) -matrix, (ai ) . (Recall that when we identify linear maps with matrices we write elements in lRn and lRm as column vectors. ) A matrix gives a linear map via matrix multiplication on the left : -- mm n This correspondence identifies d(lRn , lRm ) with - We use this identification to define a topology on , lRm) . Thus, an open set of linear maps is one with the following property. Given any 4 in ' the set with matrix representative . there is E > 0 SO that 17 every I/J with I/Jij E for every pair (i,j) is in the set. I - dij 1 < Ta utologically , d (lRn, lRm ) becomes homeomorphic to lRn-m via this identification. Consider the evaluation map e : 2 (lRn, fl) x. nnR"- lRm - _ given by e (9 ,x ) = 4 (x). If we give d(lRn , lRm ) x lRn the product - - topology, then e becomes continuous. The reason is that, in the coordinates a,xiL e is given by quadratic polynomials.