At present only about half of the book is in good enough shape to be posted online. These notes evolved from the lecture notes of a minicourse given in swisk, the sedano winter school on k theory held in sedano, spain, during the week january 2227 of 2007, and from those of a longer course given in the university of buenos aires, during the second half of 2006. What we have in mind here is that the square diagram ka k ka p. In mathematics, topological k theory is a branch of algebraic topology. Topological k theory of complex noncommutative spaces volume 152 issue 3 anthony blanc. Free topology books download ebooks online textbooks. Because the bers are nitedimensional vector spaces, vector bundles inherit a few important operations from vector spaces by applying the operations berwise.
It was founded to study vector bundles on topological spaces, by means of ideas now recognised as ktheory that were introduced by alexander grothendieck. Topics in algebraic and topological ktheory paul frank. Algebraic k theory of topological k theory to analyze k er we expect to use a localization sequence in algebraic k theory to reduce to the algebraic k theory of connective commutative salgebras, and to use the b6kstedt hsiangmadsen cyclotomic trace map to topological cyclic homology to compute these bhm. In mathematics, k theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.
Our philosophy is that algebraic k theory is the study of algebraic vector bundles under rational equivalence, semi topological k theory is the study. For a detailed introduction see introduction to topological ktheory. This leads naturally to the use of analytical tools from k theory and noncommutative geometry, such as cyclic cohomology, quantized calculus with fredholm modules and index pairings. 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. We will rst develop the tools of topological k theory. Using topological ktheory, atiyah gave a very short and elegant proof for adams theorem. This note will develop the ktheory of banach algebras, the theory of extensions of c algebras, and the operator ktheory of kasparov from scratch to its most advanced aspects. The second deep fact about topological k theory is the splitting principle which says roughly that as far as k theory is concerned, all vector bundles essentially look like direct sums of vector bundles each of whose summands is just a line bundle. A generalized cohomology theory has the data of a spectrum that defines it. Hopf invariant one a division algebra structure on rn is a continuous \multiplication map. In algebraic topology, it is a cohomology theory known as topological ktheory.
Topological k theory robert hines december 15, 2016 the idea of topological k theory is that spaces can be distinguished by the vector bundles they support. It is mainly work of ib madsen, who was a student of mine once upon a time, and his collaborators marcel b okstedt and lars hesselholt. May i would like to try to explain very roughly what topological cyclic homology is and what it is good for. An introduction to complex ktheory mit mathematics. In this thesis, the basic concepts of topological ktheory are introduced and studied using tools from. Topological k theory article about topological k theory by. What is called topological ktheory is a collection of generalized eilenbergsteenrod cohomology theories whose cocycles in degree 0 on a topological space xx may be represented by pairs of vector bundles, real or complex ones, on xx modulo a certain equivalence relation. Topological k theory 3 that was not the case with either of the previous example is that it is nontrivial that is, it is not homeomorphic to s1 r.
This article will summarize what is known about these applications and the comparison map. The idea of k theory is to learn about a space by studying the equivalence classes of. In mathematics, topological ktheory is a branch of algebraic topology. Topological ktheory robert hines december 15, 2016 the idea of topological ktheory is that spaces can be distinguished by the vector bundles they support. Moreover, it is possible to derive a substantial part of stable homotopy theory from ktheory. This volume is an introductory textbook to k theory, both algebraic and topological, and to various current research topics within the field, including kasparovs bivariant k theory, the baumconnes conjecture, the comparison between algebraic and topological k theory of topological algebras, the k theory of schemes, and the theory of dgcategories. In this paper we show that the homotopy fixed point set of k a, the topological k theory space of a, under complex conjugation is just k a, the topological k theory space of a.
A graded clifford module for cv is a z2graded vector bundle m x which is a z2graded module over cv, namely equipped with a structure. String topology is the study of algebraic and differential topological properties of spaces of paths and loops in manifolds. This theory interpolates between the algebraic k theory k top x of x and the topological k theory k xan of the analytic space xan associated to x. The idea of topological ktheory is that spaces can be distinguished by the vector bundles they support. Seminar on topological k theory winter 201819 the main subject of this seminar is the study of vector bundles on compact topological spaces.
First, recall that for k a field then a kvector bundle over a topological space x is a map vx whose. Stable real ktheory and real topological hochschild homology. In some respects ktheory is more elementary than classical homology and cohomology, and it is also more powerful for. Analogously, open topological string theory can be used to compute superpotentials for type ii string on cy3 with d branes. Let a be a banach algebra and a its complexification.
Topological ktheory of complex nc spaces anthony blanc this talk is about my phd thesis work with b. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as general k theory that were introduced by alexander grothendieck. The fact that the topological z 2 invariant belongs to ko 2pt cannot be seen from k theory only. Let nbe a semidirect product of the form z i zp where p is prime and the zpaction on zn is free away from the origin. An immediate corollary is that topological k theory vanishes for the radical of a the intersection of all the maximal ideals, and thus for purposes of studying topological k theory, it is no loss of generality to assume that a is semisimple, or even that a is a c algebra. In algebra and algebraic geometry, it is referred to as algebraic ktheory. Bulk and boundary invariants for complex topological. For further information or to download the part of the book that is written, go to the download page. I have recognized that there is a theory called k theory in mathematics is used also for applications in mathematical physics.
Aderemi kuku, introduction to ktheory and some applications. Topological ktheory, for masters students tilman bauer. Varvara karpova, complex topological ktheory, 2009. The way to say the relation is there is a map of spectra between algebraic k theory spectrum of complex numbers to the spectrum that defines complex topological k theory. K theory of quadratic forms, alias hermitian k theory. Topological ktheory of affine hecke algebras 3 to construct an a ne hecke algebra, we use a root datum rin a lattice x. Topological ktheory, lecture 1 matan prasma march 2, 2015 1 motivation. Pdf topological insulators and ktheory researchgate. Topological k theory, lecture 3 matan prasma march 2, 2015 recall that we assume throughout our base space bis connected. In algebraic l theory it is necessary to consider quadratic forms over more general rings, but only up to stable.
As parameters we take a tuple of nonzero complex numbers q q. Bulk spectrum and ktheory for infinitearea topological. The idea of k theory is to learn about a space by studying the equivalence classes of vector bundles parametrized by it. Some k theory of calgebras books also mention a little topological k theory as a background, you can see this book. Using topological k theory, atiyah gave a very short and elegant proof for adams theorem. Connections of ktheory to geometry and topology pdf. I am making some videos of k theory from topological to operator in my language chinese, if you can read chinese or have some friend help to. On the ktheory classification of topological states of matter.
Intersection theory in loop spaces, the cacti operad, string topology as field theory, a morse theoretic viewpoint, brane topology. Knowledge of the spectrum and k theory of the infinitearea system will control the spectrum and k theory of sufficiently large finite systems. We present a rigorous and fully consistent k theoretic framework for studying gapped phases of free fermions. Topological ktheory and bott periodicity diva portal. This unfinished book is intended to be a fairly short introduction to topological k theory, starting with the necessary background material on vector bundles and including also basic material on characteristic classes. Small seminar room, institute for theoretical physics. In mathematics, ktheory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. Klaus wirthmuller, vector bundles and ktheory, 2012.
Encyclopedia article about topological k theory by the free dictionary. On the k theoretic classification of topological phases. Topological and bivariant ktheory joachim cuntz springer. The goal of this paper is to introduce some of the basic ideas surrounding the theory of vector bundles and topological ktheory. Complex topological ktheory cambridge studies in advanced mathematics series by efton park. 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. Topological k theory, lecture 2 matan prasma march 2, 2015 again, we assume throughout that our base space bis connected. Topological ktheory, for masters students tilman bauers. Complex topological ktheory topological ktheory is a key tool in topology, differential geometry, and index theory, yet this is the. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. The plan is for this to be a fairly short book focusing on topological k theory and containing also the necessary background material on vector bundles and characteristic classes. Topics in algebraic and topological ktheory lecture notes.
The topological ktheory of certain crystallographic groups james f. The thom isomorphism for singular cohomology 94 16. These give a weyl group w wr and an extended a ne weyl group we xo w. Pdf topological field pattems of the yangmills theory. To demonstrate it, he wrote it on a postcard and mailed it to a colleague. K theory can be viewed as an attempt to classify vector bundles up to a coarser equiv. On k theory, groups, and topological dynamics abstract. Topological k theory of x is studied by mapping x into what we call ku and algebraic k theory by kc. Topological k theory is one of the most important invariants for noncommutative algebras equipped with a suitable topology or bornology. The plan is for this to be a fairly short book focusing on topological ktheory and containing also the necessary background material on vector bundles and characteristic classes.
The relation between finite volume ktheory and infinite volume chern numbers is only proven to begin, for the model under investigation here, for systems on hilbert space of dimension around 17. This has dramatic consequences, which we see when we discuss the adams operations below. What is called topological k theory is a fundamental construction and tool in the field of topology and homotopy theory. Topological k theory is a key tool in topology, differential geometry and index theory, yet this is the first contemporary introduction for graduate students new to the subject. It is also a fundamental tool in the field of operator algebras. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic k theory. At present only about half of the book is in good enough shape to be posted online, approximately 120 pages. We will rst develop the tools of topological k theory and once these will be su ciently. Aug 20, 2019 knowledge of the spectrum and ktheory of the infinitearea system will control the spectrum and ktheory of sufficiently large finite systems.
Comparison between algebraic and topological ktheory for. It utilizes and profits from powerful techniques in operator k theory, which from the point of view of symmetries such as time reversal, charge conjugation, and magnetic translations, is more general and natural than the topological version. In the classical theory of quadratic forms the ground ring is a eld, or a ring of integers in an algebraic number eld, and quadratic forms are classi ed up to isomorphism. This essay aims to provide a quick and accessible introduction to k theory, including how to calculate with it, and some of its additional features such as characteristic. These notes evolved from the lecture notes of a minicourse given in swisk, the sedano winter school on ktheory held in sedano, spain, during the week january 2227 of 2007, and from those of a longer course given in the university of buenos aires, during the second half of 2006. Free k theory books download ebooks online textbooks. Dec 19, 2002 download fulltext pdf download fulltext pdf download fulltext pdf topological field pattems of the yangmills theory article pdf available in physics letters b 55734. In algebraic topology, it is a cohomology theory known as topological k theory. The focus is on complex topological k theory, however.
There is existing algebraic k theory and topological k theory. Topological ktheory has become an important tool in topology. We recall its constructionandrefertowal85forthedetails. Free k theory books download ebooks online textbooks tutorials. Morse theory, homotopy theory and homological algebra. Topological ktheory of complex noncommutative spaces. This will be an expository seminar on the elements of topological k theory at a level suitable for graduate students in mathematics and physics. Survey of topological ktheory, operator ktheory, preliminaries, ktheory of crossed products, theory of extensions, kasparovs kktheory. Introduction we work in the setting of nc algebraic geometry of kontsevich. Some of their basic properties a re similar and derived with essentially. There is a similar equivalence between the category of. Survey of topological ktheory, operator ktheory, preliminaries, ktheory of crossed products, theory of extensions, kasparovs kk theory. The early work on topological ktheory is due to michael atiyah and friedrich hirzebruch. We will compute the topological ktheory of the real and complex group c algebra of and show that satis.
K v dif and k d which are formally analogue to k v and k h. Pdf a descent theorem in topological ktheory max karoubi. Topological ktheory is also such an invariant in fact, a generalized cohomology theory with a very different flavor. For topological ktheory one has the clasic 1967 harvard notes by. Topological k theory has an elementary definition but farreaching applications and results in diverse fields of pure mathematics, as well as in mathematical physics. To motivate this, we will use ktheoretic methods to prove adams theorem about the nonexistence of maps of hopf invariant one in dimensions other than n 1. Textbook or lecture notes in topological ktheory mathoverflow. The second part develops an operator algebraic approach for the study of disordered topological insulators. It will be shown that topological k theory is a periodic, generalised cohomology theory in which the cocycles have a geometric interpretation as equivalence classes of vector bundles. Rational algebraic k theory of topological k theory 2041 schwede proved in 31, theorem 4. In a similar vein, the real semi topological ktheory, written k.
We intend this survey to provide both motivation and coherence to the field of semi topological ktheory. We will compute the topological k theory of the real and complex group c algebra of and show that satis. In order to understand the mod 2 index theorem, we study the version of k theory relevant for the systems of topological insulators under study. The topological ktheory of certain crystallographic groups.
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