Learning seminar SoSe 2021, Philipps-Universit¨atMarburg, Dr. Panagiotis Konstantis

Selected topics of Friedrich Hirzebruch’s collected works

1. Hirzebruch surfaces, This is part of Hirezbruch’s PhD thesis. It is the definition of the Hirzebruch sufaces and the study of their properties. For a good overview about the important properties see [9]. For their proofs see [6].

2. Chern numbers and Pontryagin numbers, Introduction of Chern and Pontryagin numbers [11, §16.]. Proof of [11, Theorem 4.9] with Pontryagin numbers. Presentation of the remaining properties of Pontryagin numbers.

3. Oriented cobordism ring and a theorem of Thom (∗), Definition of the oriented cobordism ring, [11, §17], provide or blackbox everything which is necessary to prove [11, Theorem 18.8] and [11, Corollary 18.9]. See also [5, Zweites Kapitel, §6 and §7] 4. Multiplicative sequences and genera (∗), Introduce algebraic preliminaries for the definition of multplicative sequences. Define the L- genus, the Todd-genus and hat Aˆ-genus. [5, Erstes Kapitel, §1] see also [11, §19]. Present the properties of the Todd genus of an almost complex manifold according to [5, Drittes Kapitel, §10]. Show as a corollary the identity on [7, Band I, p. 777]. 5. Hirzebruch’s signature theorem and the L-genus, There is no need to define the signature, since I was told that everyone knows what the signature of a manifold is. The proof of the theorem is contained in [5, Zweites Kapitel, §8]. One should explicitely show Theorem 8.2.1 II).

6. Basics of topological K-theory, We review the basic properties of the Grothendieck group of the free group of real respect- ively complex vector bundles. Furtheremore we would like to remind the seminar about the Chern character and its real analogon, which establish a connection between K-theories and the cohomology groups of a topological space. Finally we will discuss without proof the Bott- periodicity. As for a reference the topics of [8, §1 and §2] should be covered. 7. The Atiyah-Singer index theorem and the Hirzebruch-Riemann-Roch theorem (∗), We will learn how to use the index theorem to compute stuff. After a short explanation of the theorem we will reprove Hirzebruch’s signature theorem using the index theorem. On top we will also show the Hirzebruch-Riemann-Roch theorem as another application of the index theorem, [12].

8. Differentiable Riemann-Roch theorem (∗), We will prove a differentiable Riemann-Roch theorem which is a generalization of the Hirzebruch- Riemann-Roch theorem and which has a bunch of interesting corollaries. For example the integrality of the Todd-genus and the Aˆ-genus can be deduced from it as well as some homotopy invariance property of Pontryagin classes. The relevant parts are [8, §3-§5] and [1]. 9. Characteristic classes and homogeneous spaces, Characteristic classes on homogeneous spaces G/H are strongly connected to the algebraic data of G and H. In this talk we would like to remind ourself about the basic properties of repres- entation theory and then study the connection to characteristic classes, [4, Chapter I, Chapter III].

10. Bott periodicity and the parallelizability of the spheres, Nowadays it is well known that the only parallizable spheres can be found in dimension 1, 3 and 7. Atiyah and Hirzebruch gave a short proof of the above fact in [2] using K-theory, the Bott periodicity theorem and a theorem of Milnor. First we discuss the theorem of Milnor [10, Theorem 2]. Instead of using the first two theorems of [10] to prove the claim, we will use [2, Theorem 1].

11. Spin-manifolds and group actions, This article of Atiyah and Hirzebruch give an obstruction for a manifold to admit an action of a compact Lie group on a spin-manifold. Namely the authors show, that if such a Lie group action exists, then the Aˆ-genus must vanish, [3]. Some theorems on index theory must be blackboxed.

References

[1] M. F. Atiyah and F. Hirzebruch. “Riemann-Roch theorems for differentiable manifolds”. Bull. Amer. Math. Soc. 65 (1959), pp. 276–281. [2] M. F. Atiyah and F. Hirzebruch. “Bott periodicity and the parallelizability of the spheres”. Proc. Cambridge Philos. Soc. 57 (1961), pp. 223–226. [3] M. Atiyah and F. Hirzebruch. “Spin-manifolds and group actions”. Essays on Topology and Related Topics (M´emoires d´edi´es`aGeorges de Rham). Springer, New York, 1970, pp. 18–28. [4] A. Borel and F. Hirzebruch. “Characteristic classes and homogeneous spaces. I”. Amer. J. Math. 80 (1958), pp. 458–538. [5] F. Hirzebruch. Neue topologische Methoden in der algebraischen Geometrie. Zweite erg¨anzteAu- flage. Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Heft 9. Springer-Verlag, Berlin- G¨ottingen-Heidelberg, 1962, pp. vii+181. [6] F. Hirzebruch. “Uber¨ eine Klasse von einfachzusammenh¨angendenkomplexen Mannigfaltigkeiten”. Math. Ann. 124 (1951), pp. 77–86. [7] F. Hirzebruch. Gesammelte Abhandlungen. Band I, II. 1951–1962, 1963–1987. Springer-Verlag, Berlin, 1987, Vol. I: viii+814 pp. Vol. II: iv+818. [8] F. Hirzebruch. “A Riemann-Roch theorem for differentiable manifolds”. S´eminaire Bourbaki, Vol. 5. Soc. Math. France, Paris, 1995, Exp. No. 177, 129–149. [9] M. Kreck and D. Crowley. “Hirzebruch sufaces” (2011). http://www.boma.mpim-bonn.mpg. de/data/25print.pdf. [10] J. Milnor. “Some consequences of a theorem of Bott”. Ann. of Math. (2) 68 (1958), pp. 444–449. [11] J. W. Milnor and J. D. Stasheff. Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974, pp. vii+331. [12] P. Shanahan. The Atiyah-Singer index theorem. Vol. 638. Lecture Notes in Mathematics. An introduction. Springer, Berlin, 1978, pp. v+224.