INTRODUCTION TO DIFFERENTIAL TOPOLOGY

1 INTRODUCTION TODIFFERENTIAL TOPOLOGYJoel W. RobbinUW MadisonDietmar A. SalamonETH Z urich14 August 2018iiPrefaceThese are notes for the lecture course DIFFERENTIAL Geometry II held by thesecond author at ETH Z urich in the spring semester of 2018. A prerequisiteis the foundational chapter about smooth manifolds in [21] as well as somebasic results about geodesics and the exponential map. For the benefit ofthe reader we summarize some of the relevant background material in thefirst chapter and in the appendix. The lecture course covered the content ofChapters 1 to 7 (except Section ).The first half of this book deals with degree theory and the Pointar e Hopftheorem, the Pontryagin construction, intersection theory, and Lefschetznumbers.

2 In this part we follow closely the beautiful exposition of Milnorin [14]. For the additional material on intersection theory and Lefschetznumbers a useful reference is the book by Guillemin and Pollack [9].The second half of this book is devoted to DIFFERENTIAL forms and de Rhamcohomology. It begins with an elemtary INTRODUCTION into the subject andcontinues with some deeper results such as Poincar e duality, the Cech deRham complex, and the Thom isomorphism theorem. Many of our proofsin this part are taken from the classical textbook of Bott and Tu [2] whichis also a highly recommended reference for a deeper study of the subject(including sheaf theory, homotopy theory, and characteristic classes).

3 14 August 2018 joel W. Robbin and Dietmar A. SalamoniiiivContentsIntroduction11 Degree Theory Modulo Smooth Manifolds and Smooth Maps .. The Theorem of Sard and Brown .. Manifolds with Boundary .. Proof of Sard s Theorem .. The Degree Modulo Two of a Smooth Map .. The Borsuk Ulam Theorem .. 292 The Brouwer Oriented Manifolds and the Brouwer Degree .. Zeros of a Vector Field .. Zeros .. Zeros .. The Poincar e Hopf Theorem .. 333 Homotopy and Framed The Pontryagin Construction .. The Product Neighborhood Theorem .. The Hopf Degree Theorem .. 354 Intersection Transversality .. Intersection Numbers.

4 Numbers Modulo Two .. and Intersection Numbers .. Intersections .. Self-Intersection Numbers .. The Lefschetz Number of a Smooth Map .. 69vviCONTENTS5 DIFFERENTIAL Exterior Algebra .. Forms .. Product and Pullback .. Forms on Manifolds .. The Exterior DIFFERENTIAL and Integration .. Exterior DIFFERENTIAL on Euclidean Space .. Exterior DIFFERENTIAL on Manifolds .. Theorem of Stokes .. The Lie Derivative .. s Formula .. and Exactness .. Volume Forms .. and Degree .. Gau Bonnet Formula .. Isotopy .. 1156 De Rham The Poincar e Lemma .. The Mayer Vietoris Sequence .. Exact Sequences.

5 Good Covers .. K unneth Formula .. Compactly Supported DIFFERENTIAL Forms .. and Basic Properties .. Mayer Vietoris Sequence forH c.. Poincar e Duality .. Poincar e Pairing .. of Poincar e Duality .. e Duality and Intersection Numbers .. Characteristic and Betti Numbers .. and Exercises .. The Cech de Rham Complex .. Cech Complex .. Isomorphism .. Cech de Rham Complex .. Structures .. on De Rham s Theorem .. 174 CONTENTSvii7 Vector Bundles and the Euler Vector Bundles .. The Thom Class .. over the Fiber .. Thom Isomorphism Theorem .. Theory Revisited .. The Euler Class .. Euler Number .. Euler Class.

6 Product Structure onH (CPn) .. 2108 Connections and Connections .. Valued DIFFERENTIAL Forms .. Transport .. Groups .. Connections .. Curvature .. and basic properties .. Bianchi Identity .. Transformations .. Connections .. Chern Weil Theory .. Polynomials .. Classes .. Euler Class of an Oriented Rank-2 Bundle .. Examples .. Chern Classes .. and Properties .. of the Chern Classes .. of Existence and Uniqueness .. Products of Complex Line Bundles .. Chern Classes in Geometry .. Manifolds .. Adjunction Formula .. Surfaces .. Complex Structures on Four-Manifolds .. Low-Dimensional Manifolds.

7 268viiiCONTENTSA Paracompactness .. Partitions of Unity .. Embedding a Manifold into Euclidean Space .. Riemannian Metrics .. The Exponential Map .. Classifying Smooth One-Manifolds .. 291 References293 Index294 Introduction12 CONTENTSC hapter 1 Degree Theory Modulo TwoIn this and the following two chapters we follow closely the beautiful book TOPOLOGY from the Differentiable Viewpoint by Milnor [14]. Milnor s mas-terpiece of mathematical exposition cannot be improved. The only excusewe can offer for including the material in this book is for completeness ofthe exposition. There are, nevertheless, two minor points in which the firstthree chapters of this book differ from [14].

8 The first is that our expositionuses the intrinsic notion of a smooth manifold. The basic definitions areincluded in Section and the proofs of some foundational theorems suchas the existence of partitions of unity and of embeddings in Euclidean spaceare relegated to the appendix. For a more extensive discussion of these con-cepts the reader is referred to the two introductory chapters of [21] whichare understood as prerequisites for the present book. A second minor pointof departure from Milnor s text is the inclusion of the Borsuk Ulam theoremin Section at the end of the present chapter. The other four section ofthis chapter correspond to the first four chapters of Milnor s book. Afterthe introductory section, which includes a proof of the fundamental theo-rem of algebra, we discuss Sard s theorem, manifolds with boundary, andthe Brouwer Fixed Point Theorem in Section , include a proof of Sard sTheorem in Section , and introduce the degree modulo two of a smoothmap in Section Throughout we assume that the reader is familiar withfirst year analysis and the basic notions of point set 1.

9 DEGREE THEORY MODULO Smooth Manifolds and Smooth MapsLetU RmandV Rnbe open sets. A mapf:U Vis calledsmoothiff it is infinitely differentiable, iff all its partial derivatives f= 1+ + mf x 11 x mm, = ( 1,.., m) Nm0,exist and are continuous. For a smooth mapf= (f1,..,fn) :U Vanda pointx Uthederivative offatxis the linear mapdf(x) :Rm Rndefined bydf(x) :=ddt t=0f(x+t ) = limt 0f(x+t ) f(x)t, linear map is represented by theJacobian matrixoffatxwhichwill also be denoted bydf(x) := f1 x1(x) f1 xm(x).. fn x1(x) fn xm(x) Rn that we use the same notation for the Jacobian matrix and the cor-responding linear map fromRmtoRn. The derivative satisfies thechainrule. Namely, ifU Rm,V Rn,W Rpare open sets andf:U Vandg:V Ware smooth maps theng f:U Wis smooth andd(g f)(x) =dg(f(x)) df(x) :Rm Rp( )for everyx U.

10 Moreover the identity map idU:U Uis always smoothand its derivative at every point is the identity map ofRm. This impliesthat, iff:U Vis adiffeomorphism( bijective andfandf 1are both smooth), then its derivative at every point is an invertible linearmap and som=n. The Inverse Function Theorem is a partial converse (seeTheorem below for maps between manifolds).Following Milnor [14], we extend the definition of smooth map to mapsbetween subsetsX RmandY Rnwhich are not necessarily open. Inthis case a mapf:X Yis calledsmoothif for eachx0 Xthere existsan open neighborhoodU Rmofx0and a smooth mapF:U Rnthatagrees withfonU X. A mapf:X Yis called adiffeomorphismiffis bijective andfandf 1are smooth. When there exists a diffeomor-phismf:X YthenXandYare open these definitions coincide with the usage SMOOTH MANIFOLDS AND SMOOTH MAPS5 Smooth ManifoldsDefinition (Smoothm-Manifold).