The standard texts on the subject are by michael atiyah and max karoubi, both called ktheory,i believe. Pages 46 4 letter hirzebruch atiyah dated october 7 1958. I am looking for a good introductory level textbook or set of lecture notes on classical topological k theory that would be suitable for a onesemester graduate course. These are notes for a talk given in the kan seminar at mit in fall 2015. What follows is an edited version of those catchascatchcan conversations.
Geometric khomology and the atiyahsinger index theorem. Atiyah and singer then used it to great e ect when they proved their theorem on the index of elliptic operators. Lectures on k theory pdf 95p this lecture note covers the following topics. Also, projective modules are treated below, but not in their book. Several results showed that the newly introduced k theory was in some ways more powerful than ordinary cohomology theory. Michael atiyahs mathematical dreams quanta magazine. Solutions to atiyah and macdonalds introduction to. A brief account of k theory written in honour of friedrich hirzebruch.
Page 3 3 letter atiyah hirzebruch dated september 29 1958. Geometric, algebraic, and arithmetic objects are assigned objects called kgroups. Localization, whitehead groups and the atiyah conjecture. The theme of my talk is a generalization of this duality motivated by orbifold superstring theory of type iib, in which principal circle bundles are replaced by. We define generalized atiyahpatodisinger boundary conditions of product type for dirac operators associated to cvector bundles on the product of a compact manifold with boundary and a closed manifold. This monoid is known to be realizable by the algebra a kf of the monogenic. Product formula for atiyahpatodisinger index classes and.
I am looking for a good introductory level textbook or set of lecture notes on classical topological ktheory that would be suitable for a onesemester graduate course. So far, this conjecture has almost exclusively been stated and proved for compact lie groups g acting on compact symplectic manifolds, and, largely due to the use of spin c dirac operator techniques, has reached a high degree of perfection under these. Further development of atiyahsinger index theorem and ktheory date. We describe a variant of ktheory for spaces with involution, built from vector bundles which are sent to their negative under the involution. In this talk, i give the proof of the completion theorem in equivariant k theory given by atiyah and segal in. In order to define kx, one considers first the free abelian group lx generated. The course is about the atiyah singer index theorem. Now that hes officially presented it, is atiyahs proof. The atiyahsinger index theorem encompasses many other important results. Repeating the construction, we obtain k 2 in which every f has two roots if possible, and similarly we obtain k nfor al n. Topological theory has become an important tool in topology. In mathematics, k theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. The guilleminsternberg conjecture states that quantisation commutes with reduction in a specific technical setting.
This theory was invented by alexander grothendieck1 bs in the 50s. Michael atiyah these notes are based on the course of lectures i gave at harvard in the fall of 1964. It is this topological jtheory that this book will study. We define topological and analytic indices in rz k theory and show that they are equal. In algebraic topology, it is a cohomology theory known as topological ktheory. Geometric, algebraic, and arithmetic objects are assigned objects called k groups. Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device. Adams and atiyah used ktheory to reprove the above resul ts concerning which spheres are mod 2 hspaces.
We explore the relations of twisted ktheory to twisted and untwisted classical cohomology. We prove a product formula for the ktheoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures. Your print orders will be fulfilled, even in these challenging times. In order to motivate this theory, let us first recall the integral analog, the atiyahsinger families index theorem. It is also a fundamental tool in the field of operator algebras. Atiyah and hirzebruch considered a topological analog defined for any. Ktheory by michael atiyah, 9780201407921, available at book depository with free delivery worldwide. Free k theory books download ebooks online textbooks.
The atiyah book is more readable and has fewer prerequisites,but the karoubi book covers a great deal more. You are especially wellknown for two supremely beautiful works, not only the index theorem but also ktheory, developed with the german topologist friedrich hirzebruch. Mar 03, 2016 quanta magazine sat down with atiyah at the royal society gathering and afterward, whenever he slowed down long enough to take questions. Textbook or lecture notes in topological ktheory mathoverflow. In fact, rational cohomology is defined in terms of k theory. Product formula for atiyahpatodisinger index classes and higher signatures volume 6 issue 2 charlotte wahl.
In fact, rational cohomology is defined in terms of ktheory. The early development of topological ktheory by michael atiyah and. We construct an atiyah hirzebruch spectral sequence, and describe. We describe a variant of k theory for spaces with involution, built from vector bundles which are sent to their negative under the involution. Let c be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. From minimal geodesics to susy field theories 3 the sequence e n, n n. Using k theory, adams and atiyah were able to give a simple proof that the only spheres which can be provided with hspace structures are s1, s3 and s7. He is best known for his work on the ktheory and the atiyahsinger index theorem. Rz index theory university of california, berkeley. Introduction to ktheory and some applications math ksu.
In order to motivate this theory, let us first recall the integral analog, the atiyah singer families index theorem. Enter your mobile number or email address below and well send you a link to download the free kindle app. An elementary introduction by max karoubi conference at the clay mathematics research academy the purpose of these notes is to give a feeling of ktheory, a new interdisciplinary subject within mathematics. Blackadars book for ktheory of operator algebras, and atiyahs book for topological ktheory as it stood in the 1960s, are readable without a lot of algebraic prerequisites. In this chapter we will describe restrictions obtained for torsion free mod 2 finite hspaces via ktheory. The proposal that dbrane charge on orbifolds is given by equivariant ktheory goes back to. We will describe an extension of the index formula for closed manifolds see atiyah and singer 1963 to manifolds with boundary.
Its statement is given below, although it may not make sense at the moment. Atiyah and hirzebruch applied this construction to vector bundles locally free sheaves over a compact manifold m, creating topological k theory. Ktheory is a tool for working with higherdimensional analogues of this example, or in other words for describing higherdimensional twistings. Moreover, it is possible to derive a substantial part of stable homotopy theory from ktheory. In the present book, category theory is a basic tool. Where do you trace the beginnings of your interest in beauty and science. Grothendieck, atiyah and hirzebruch in order to formalize his work on the riemannroch theorem in the spirit of hirzebruch, grothendieck introduced a new contravariant functor bs defined on the category of non singular algebraic varieties x. A classical application of ktheory is to the problem of the existence of vector elds on spheres, solved by adams. They constitute a selfcontained account of vector bundles and ktheory assuming only the rudiments of pointset topology and linear algebra. The theory is taken as far as the solution of the hopf invariant problem and a start is mode on the jhomomorphism. Equivariant k theory and completion siddharth venkatesh abstract. In mathematics, ktheory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.
Topology is a branch of ktheory and ktheory is a branch of topology and i had. They contain detailed information about the original object but are notoriously difficult to compute. Pdf on nov 1, 2019, alain connes and others published sir michael atiyah, a knight mathematician find, read and cite all the research you need. This idea of k theory has invaded other parts of mathematics, for example number theory ga, topology bott ah1 and functional analysis connes. Correspondence atiyah hirzebruch about ktheory abstract 1 letter atiyah hirzebruch dated december 31 1957. Ktheory crc press book these notes are based on the course of lectures i gave at harvard in the fall of 1964. The index of a dirac operator d of a clifford module e over a compact, oriented manifold m of even dimension is indd. The guilleminsternberg conjecture for noncompact groups. I am free, and could come at any suitable time to synchronize with other people. Topology is a branch of k theory and k theory is a branch of topology and i had. An atiyahhirzebruch spectral sequence for krtheory 3 gets s2,1. An atiyahhirzebruch spectral sequence for krtheory arxiv.
Algebraic ktheory, second edition, birkhauser, 1996. The ktheoretic proof of the nonexistence of elements of hopf invariant one note that this uses adams operations, see e. Introduction the purpose of this paper is to introduce an index theory in which the indices take value in rz. Since equivariant ktheory on a space with free gaction is the same as the ktheory of the quotient, we have the following consequence of the completion theorem. Page 2 2 letter atiyah hirzebruch dated september 8 1958. He is best known for his work on the k theory and the atiyah singer index theorem. With x a point, the original atiyahsegal completion theorem specializes to a calculation of the ktheory of classifying spaces in terms of completions of representation rings. Atiyah received 9 august 1966 introduction the theory of complex vector bundles 2, 5 has many variants and refinements. Thus they discuss the universal mapping property ump of localization of a ring, but provide an ad hoc. Walker 674 notices of the ams volume 53, number 6 sir michael atiyah, winner of both a fields medal and an abel prize, delivered the first annual einstein public lecture at the university of nebraskalincoln. A geometric introduction to ktheory daniel dugger contents preface 3 introduction 4 1. A brief account of ktheory written in honour of friedrich hirzebruch subjects.
Algebraic k theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. The aim of these lectures is to report on the progress of the index problem in the last year. The heat equation and the atiyahsinger index theorem. Pdf on nov 1, 2019, alain connes and others published sir michael atiyah, a knight mathematician find, read and cite all the research you need on researchgate. If you dont want to wait have a look at our ebook offers and start reading immediately. In algebra and algebraic geometry, it is referred to as algebraic ktheory. Atiyah and todd used k theory to improve the lower bounds found using ordinary cohomology by borel and serre for the james number, describing when a map from a complex stiefel manifold to a. Atiyah shows in an earlier paper that there is some gspace y such that k0 g x is. Among many successes of k theory, one should mention the solution of classical topological problems k chapter v, the atiyah. A geometric introduction to ktheory university of oregon.
Free k theory books download ebooks online textbooks tutorials. Moreover, it is possible to derive a substantial part of stable homotopy theory from k theory. These notes are based on the course of lectures i gave at harvard in the fall of 1964. And is there material lecture video or good pdf script where the algebraic ktheory is explained. Atiyah and hirzebruch applied this construction to vector bundles locally free sheaves over a compact manifold m, creating topological ktheory. Further development of atiyahsinger index theorem and ktheory. A brief account of ktheory written in honour of friedrich hirzebruch. An elementary introduction by max karoubi conference at the clay mathematics research academy the purpose of these notes is to give a feeling of k theory, a new interdisciplinary subject within mathematics. An elementary introduction by max karoubi clay mathematics academy the purpose of these notes is to give a feeling of ktheory, a new interdisciplinary subject within mathematics. In algebraic topology, it is a cohomology theory known as topological k theory. Check our section of free ebooks and guides on k theory now. The ktheory and the index theorem are studied in atiyahs book ktheory 1967, reprinted 1989 and his joint work with g b segal, the index of elliptic operators iv, in the annals of mathematics, volumes 88 and 93 1968, 1971. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end.
They constitute a selfcontained account of vector bundles and k theory assuming only the rudiments of pointset topology and linear algebra. Pdf free lamplighter groups and a question of atiyah. The analogs of vector bundles are locally free coherent sheaves, or just finitelygenerated projective. One of the features of the treatment is that no use is made of ordinary. Topological ktheory has become an important tool in topology. Now that hes officially presented it, is atiyahs proof of. With x a point, the original atiyah segal completion theorem specializes to a calculation of the k theory of classifying spaces in terms of completions of representation rings.
Solutions to atiyah and macdonalds introduction to commutative algebra athanasios papaioannou august 5, 2004. Further development of atiyahsinger index theorem and k. Algebraic ktheory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Atiyah and hirzebruch 3 con sidered a topological analog defined for any compact space x, a group kx constructed from the category of vector bundles on x. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory.
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