Over the years, I have taught several courses on dierential topology in the master's degree program in mathematics at the University of Pisa. The class was usually attended by students who had accomplished (or were accomplishing) a rst three years degree in mathematics, together with a few peer physicists and a few beginner Ph.D. students. Considering the initial knowledge of these students, time after time, a collection of dierent topics, in dierent combinations, as well as a certain way to present them, has emerged. This textbook summarizes such teaching experiences, therefore it presents itself more as lecture notes" than as a complete and systematic treatise. Sometimes, in a class, a short cut" to an interesting application is chosen over broader generality. Similarly, in this text we will focus, for example, on compact manifolds (especially when we consider the sources of smooth maps), allowing simplications in dealing, for instance, with function spaces or with certain globalization procedure" of maps. There are already a lot of interesting facts concerning compact manifolds, so we will do it without remorse. There are several classical well-known references (such as [M1], [GP], [H], [M2], [M3], [Mu], : : : ) which I used in preparing the courses and which have strongly in uenced these pages. So, why another textbook on dierential topology? An important motivation came to me from the students, looking at their notes and from their remark that hey had not been able to nd some of the topics addressed in the course anywhere". It would be very hard to claim any `originality' in dealing with such a classical matter. However, that remark, at least in reference to textbooks addressed mainly to undergraduate readers, has some truth to it. Let's make an example. A theme of this text (similarly, for example, to [H]) is the synergy between bordism and transversality. One of the limits imposed by the students' presumed initial knowledge, as mentioned before, is that we can't assume any familiarity with algebraic topology or homological algebra (besides, perhaps, the very basic facts about homotopy groups); on the other hand, it is very useful and meaningful to dispose of a (co)-homology theory suited to support several dierential topology constructions. We will show that (oriented or non-oriented) bordism provides instances of so-called (covariant) generalized" homology theories for arbitrary pairs (X;A) of topological spaces, constructed via geometric means. Then, by specializing X to be a smooth compact manifold, and after a re-idexing of the bordism modules by the codimension (so that they are now called cobordism modules), transversality allows to incorporate the bordism modules into a contravariant cobordism functor with the category of graded rings as the target; the product on cobordism modules is also dened by direct geometric means. This multiplicative structure is a substantial enhancement and it will lead to several important and often very classical applications. For example, it is the natural context for unavoidable topics such as the degree theory or the Poincare-Hopf index theorem. The verication that several constructions are well-dened is eventually reduced to the fact that the cobordism product is well-dened. Moreover, when possible, the invariance up to bordism" is emphasized rather than the invariance up to homotopy", compared to most of the established references. Not assuming any familiarity with algebraic topology, this presentation could also be useful as an intuitive, geometrically based introduction to some topics of that discipline. Overall, this book is a collection of themes, in some cases advanced and of historical importance and whose choice was certainly due in part to personal preferences, with the common characteristic that they can be treated with are hands", meaning by combining specic dierential-topological cut-and-paste procedures and applications of transversality, mainly through the cobordism multiplicative structure. The trait of geometric construction sets the one" of this textbook, intended to be accessible and useful to motivated master undergraduate students and Ph.D. students, but also to a more expert reader to recognize very basic reasons for some facts already known as the result of more advanced theories or technologies.
Lectures on differential topology
Riccardo Benedetti
2021-01-01
Abstract
Over the years, I have taught several courses on dierential topology in the master's degree program in mathematics at the University of Pisa. The class was usually attended by students who had accomplished (or were accomplishing) a rst three years degree in mathematics, together with a few peer physicists and a few beginner Ph.D. students. Considering the initial knowledge of these students, time after time, a collection of dierent topics, in dierent combinations, as well as a certain way to present them, has emerged. This textbook summarizes such teaching experiences, therefore it presents itself more as lecture notes" than as a complete and systematic treatise. Sometimes, in a class, a short cut" to an interesting application is chosen over broader generality. Similarly, in this text we will focus, for example, on compact manifolds (especially when we consider the sources of smooth maps), allowing simplications in dealing, for instance, with function spaces or with certain globalization procedure" of maps. There are already a lot of interesting facts concerning compact manifolds, so we will do it without remorse. There are several classical well-known references (such as [M1], [GP], [H], [M2], [M3], [Mu], : : : ) which I used in preparing the courses and which have strongly in uenced these pages. So, why another textbook on dierential topology? An important motivation came to me from the students, looking at their notes and from their remark that hey had not been able to nd some of the topics addressed in the course anywhere". It would be very hard to claim any `originality' in dealing with such a classical matter. However, that remark, at least in reference to textbooks addressed mainly to undergraduate readers, has some truth to it. Let's make an example. A theme of this text (similarly, for example, to [H]) is the synergy between bordism and transversality. One of the limits imposed by the students' presumed initial knowledge, as mentioned before, is that we can't assume any familiarity with algebraic topology or homological algebra (besides, perhaps, the very basic facts about homotopy groups); on the other hand, it is very useful and meaningful to dispose of a (co)-homology theory suited to support several dierential topology constructions. We will show that (oriented or non-oriented) bordism provides instances of so-called (covariant) generalized" homology theories for arbitrary pairs (X;A) of topological spaces, constructed via geometric means. Then, by specializing X to be a smooth compact manifold, and after a re-idexing of the bordism modules by the codimension (so that they are now called cobordism modules), transversality allows to incorporate the bordism modules into a contravariant cobordism functor with the category of graded rings as the target; the product on cobordism modules is also dened by direct geometric means. This multiplicative structure is a substantial enhancement and it will lead to several important and often very classical applications. For example, it is the natural context for unavoidable topics such as the degree theory or the Poincare-Hopf index theorem. The verication that several constructions are well-dened is eventually reduced to the fact that the cobordism product is well-dened. Moreover, when possible, the invariance up to bordism" is emphasized rather than the invariance up to homotopy", compared to most of the established references. Not assuming any familiarity with algebraic topology, this presentation could also be useful as an intuitive, geometrically based introduction to some topics of that discipline. Overall, this book is a collection of themes, in some cases advanced and of historical importance and whose choice was certainly due in part to personal preferences, with the common characteristic that they can be treated with are hands", meaning by combining specic dierential-topological cut-and-paste procedures and applications of transversality, mainly through the cobordism multiplicative structure. The trait of geometric construction sets the one" of this textbook, intended to be accessible and useful to motivated master undergraduate students and Ph.D. students, but also to a more expert reader to recognize very basic reasons for some facts already known as the result of more advanced theories or technologies.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
