Hatcher, ktheory and vector bundles, pdf version 2. 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. What sets ktheory apart from other cohomology theories is that it can be defined quite directly by vector. The use of ktheory to connect with the invariants of volovik 2 and essingurarie 64 is new. This theory interpolates between the algebraic ktheory k top x of x and the topological. In this course we will study define and study topological ktheory. Lectures on k theory pdf 95p this lecture note covers the following topics. 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. In algebra and algebraic geometry, it is referred to as algebraic ktheory. In mathematics, ktheory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. Consideration of these questions led waldhausen to introduced his algebraic ktheory of spaces.
Topological k theory of complex noncommutative spaces volume 152 issue 3 anthony blanc. Some k theory of calgebras books also mention a little topological k theory as a background, you can see this book. Complex topological ktheory cambridge studies in advanced mathematics book 111 kindle edition by park, efton. My aim here is to present it as such and develop some of the key structures of ktheory as a cohomology theory. An introduction to complex ktheory mit mathematics.
No background in algebraic topology is assumed, every aspect of the topic is covered. Complex topological ktheory cambridge studies in advanced. Complex vector bundles are the rst main ingredient one would need to understand complex topological k theory, so chapter 3 is entirely devoted to their study. Nov 30, 2012 the topological realization is the left kan extension of the functor space of complex points to all simplicial presheaves over complex algebraic varieties. 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. The publishers may very well be exactly right in their backcover description of the text as the definitive book for a first course. Topological ktheory of complex nc spaces anthony blanc this talk is about my phd thesis work with b. 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. This ring of virtual bundles is our object of study. The goal of this project is to understand one such family of groups, collectively called topological ktheory. The semi topological ktheory of a complex variety was defined in fw2 with the expectation that it would prove to be a theory lying part way between the algebraic ktheory of the variety. 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. We compute an explicit expression for the plethory of operations for complex topological ktheory.
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. From ktheory to physics emil prodan, hermann schulzbaldes auth. The plethory of operations in complex topological ktheory. This theory is given as the homotopy groups of an infinite loop space k semi x which is equipped with maps k alg x k semi x, k semi x k top x an whose composition is the natural map from the algebraic k. This monograph offers an overview of rigorous results on fermionic topological insulators from the complex classes, namely, those without symmetries or with just a. Stacey and whitehouse showed this structure is present on the cohomology operations for a suitable generalised cohomology theory. A textbook account of topological ktheory with an eye towards operator ktheory is section 1 of. 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. Topological ktheory of complex noncommutative spaces. In mathematics, k theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. Show full abstract locally trivial, real, or complex vector.
Project description the aim of algebraic topology is to understand a topological space by associating to it groups whose algebraic properties re ect the topological properties of the space. Download it once and read it on your kindle device, pc, phones or tablets. Core level spectroscopy of solids advances in condensed matter science cubase sxsl 2 power. Free k theory books download ebooks online textbooks tutorials. Because the bers are nitedimensional vector spaces, vector bundles inherit a. The algebraic ktheory of m is a space am which is defined so that it plays essentially the same role for higher kgroups as k 1 z. New results include a generalized streda formula and a proof of the delocalized nature of surface states in topological insulators with nontrivial invariants. Survey of topological ktheory, operator ktheory, preliminaries, ktheory of crossed products, theory of extensions, kasparovs kktheory. Another natural example is the topological ktheory of a compact space x, which is the ktheory of the category vect x of. Making the effort to write the report in english was a pleasure for me. An extraordinary cohomology theory sati es all of the above except the dimension axiom. We can also define complex vector bundles in exactly the same way. Topological ktheory 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.
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. Complex topological ktheory topological ktheory is a key tool in topology, differential geometry, and index theory, yet this is the. This monograph offers an overview of rigorous results on fermionic topological insulators from the complex classes, namely, those without symmetries or with just a chiral symmetry. A generalized cohomology theory has the data of a spectrum that defines it. Comparison between algebraic and topological ktheory for. Jan 06, 2020 we provide a concrete introduction to the topologised, graded analogue of an algebraic structure known as a plethory, originally due to tall and wraith. Complex topological ktheory topological ktheory 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. Hopf invariant one a division algebra structure on rn is a continuous \multiplication map. Complex topological k theory cambridge studies in advanced mathematics series by efton park. 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. In mathematics, topological ktheory is a branch of algebraic topology. 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.
The ring of isomophism classes of complex virtual bundles over x is the k theory of. Complex topological k 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. Complex topological k theory cambridge studies in advanced mathematics conscientious objection. In algebraic topology, it is a cohomology theory known as topological ktheory. Particular focus is on the stability of the topological invariants in the presence of strong disorder, on the interplay between. Kathryn hess bellwald, for helping me all through the semester to improve my understanding of the subject. What is called topological ktheory is a collection of generalized. From k theory to physics emil prodan, hermann schulzbaldes auth. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Varvara karpova, complex topological ktheory, 2009. In algebraic topology, it is a cohomology theory known as topological k theory. This theory interpolates between the algebraic ktheory k. What we have in mind here is that the square diagram ka k ka p. Electromagnetic coupling on complex systems topological approach. In mathematics, topological k theory is a branch of algebraic topology. It was discovered, starting in the early 80s, that the \comparison map from algebraic to topological ktheory is a surprisingly rich object. Efton park topological ktheory is a key tool in topology, differential geometry and index theory, yet this is the first contemporary introduction for graduate students new. We provide a concrete introduction to the topologised, graded analogue of an algebraic structure known as a plethory, originally due to tall and wraith. From ktheory to physics this monograph offers an overview on the topological invariants in fermionic topological. X of x and the topological ktheory k top x an of the analytic space x an associated to x, in the sense that there are natural maps k x k semi x k top x an of infinite loop spaces cf. Complex topological ktheory by efton park overdrive. We will compute the topological ktheory of the real and complex group c algebra of and show that satis. The topological realization is the left kan extension of the functor space of complex points to all simplicial presheaves over complex algebraic.
Use features like bookmarks, note taking and highlighting while reading complex topological ktheory cambridge studies in advanced mathematics book 111. The topological ktheory of certain crystallographic groups. Some ktheory of calgebras books also mention a little topological ktheory as a background, you can see this book. Yes, complex topological ktheory is wellwritten and pitched at the right level for the motivated reader. 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 topological ktheory of certain crystallographic groups james f. Bulk and boundary invariants for complex topological insulators. The definition of topological ktheory consists in two steps. Connections of ktheory to geometry and topology pdf. Our first main result states that the topological k theory of the unit dgcategory is the spectrum bu. 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. Park has included some very interesting material, making this book a terrific introduction to ktheory.
The goal of this project is to understand one such family of groups, collectively called topological k theory. Introduction we work in the setting of nc algebraic geometry of kontsevich and al. Complex vector bundles are the rst main ingredient one would need to understand complex topological ktheory, so chapter 3 is entirely devoted to their study. Free k theory books download ebooks online textbooks. It is also a fundamental tool in the field of operator algebras. Complex ktheory was one of the rst extraordinary cohomology theories to be discovered and studied in depth.
These notes give an introduction to topological k theory, of the basic definitions and results, for readers with a basic knowledge of topology. 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. So a nc spacec is a dgcategoryc lets begin with a classical picture. Pdf file of the complete article 296k, or click on a page image below to browse page by page. Complex topological ktheory cambridge studies in advanced mathematics conscientious objection. Aderemi kuku, introduction to ktheory and some applications. These notes give an introduction to topological ktheory, of the basic definitions and results, for readers with a basic knowledge of topology. The internal tensor product of vector bundles induces products in these cohomology theories. The aim of algebraic topology is to understand a topological space x by associating to it groups. With all this extra structure, ktheory becomes a powerful tool, in some ways more powerful even than ordinary cohomology. Although most of our discussion of semi topological ktheory applies only to complex varieties, we dene ksemi 0 x for x a quasiprojective variety over an arbitrary eld. Textbook or lecture notes in topological ktheory mathoverflow.
Bulk and boundary invariants for complex topological insulators from ktheory to physics july 18, 2019. 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. Request pdf bulk and boundary invariants for complex topological insulators. The material covered is largely that also found in hatcher or wirthmuller 12. Klaus wirthmuller, vector bundles and ktheory, 2012. The prime example of this is the very simple proof, once the basic machinery of complex ktheory has been set up, of the theorem that there are no. Topological ktheory 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. Complex topological ktheory mathematical association of. We compute an explicit expression for the plethory of operations for complex topological k theory. The topological realization is the left kan extension of the functor space of complex points to all simplicial presheaves over complex algebraic varieties. Bulk and boundary invariants for complex topological. This essay aims to provide a quick and accessible introduction to ktheory, including how to calculate with it, and some of its additional features such as characteristic. Electromagnetic coupling on complex systems topological.
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