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Andrius Kulikauskas
 m a t h 4 w i s d o m  g m a i l
 +370 607 27 665
 My work is in the Public Domain for all to share freely.
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Investigation: Understand Ktheory and KOtheory as relevant for Bott periodicity.
Ktheory
 In what sense is a tableaux the square root of a matrix?
 What does it mean to have a trivial curl?
 What does that say about the Laplacian?
Ktheory group {$K(X)$}
 The Ktheory group, {$K(X)$}, of a compact Hausdorff topological space is defined as the abelian group generated by isomorphism classes {$[E]$} of complex vector bundles modulo the relation that, whenever we have an exact sequence {$ 0\to A\to B\to C\to 0$} then {$[B]=[A]+[C]$} in topological Ktheory.
Ideas
 The study of a ring generated by vector bundles over a topological space or scheme.
 Involves the construction of families of Kfunctors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes.
 Can consider complex vector bundles and real vector bundles
Ktheory goals
 How dimensions are related
 Cohomology  sequence of groups  which may not exist. Motivic cohomology
 Ktheory was used for the first proof of the AtiyahSinger index theorem. They reduced the index theorem to the case of spheres, and then to the case of a point.
Ktheory ideas
 When you move from Ktheory to KOtheory, you add charge conjugation operator J.
 Circle (topology) = Complex number (rotation) (algebra)
 Husemoller? Chapter 2 on Fibre Sequences contains examples in Sections 2.5 and 2.6. Seems relevant for mastering Bott periodicity. Mentions the fibration for orthogonal groups.
Resources
Ktheory
Ktheory and complex Bott periodicity
 Matthew Magill. Topological Ktheory and Bott Periodicity A long and detailed proof of complex Bott periodicity in terms of vector bundles, KGroup, category theory and Morse theory. Deduces a homotopy equivalence of the {$K$}representative, {$\mathbf{B}U\times\mathbb{Z}$}, with its own second loop space. Then considers Ktheory as a cohomological theory and Bott periodicity's role in that.
 Peter May. A Concise Course in Algebraic Topology. Chapter 24. An Introduction to KTheory. Has a section on the complex Bott periodicity.
 Allen Hatcher. Vector Bundles & KTheory. Welcome.
 Zachary Halladay. Bott periodicity and Ktheory Considers the complex case {$\pi_n(U)=\pi_{n+2}(U)$}, based on the homotopy equivalences {$\beta :BU\rightarrow\Sigma^2BU$} or its adjoint {$\beta^*:BU\wedge S^2\rightarrow BU$}, and relates that to the Ktheoretic version: the homomorphism on reduced Ktheory {$\tilde{K}(X)\rightarrow\tilde{K}(\Sigma^2X)$} induced by taking the external tensor product of stable equivalence classes of vector bundles over {$X$} with the canonical line bundle over {$S^2$} is an isomorphism. Halladay works out the details of the explicit homotopy given by Bott in his 1959 paper, which makes the two forms compatible.
 David Manuel Murrugarra Tomairo. Bott Periodicity. Master's thesis.
 One formulation of Bott periodicity describes the twofold loop space, {$\Omega ^{2}BU$} of {$BU$}. Here, {$\Omega$} is the loop space functor, right adjoint to suspension and left adjoint to the classifying space construction. Bott periodicity states that this double loop space is essentially {$BU$} again.
 Two types of results: Periodicity of the homotopy groups of the unitary groups. Periodic theorem of vector bundles.
 Jacob Lurie: Brauer Groups in Stable Homotopy Theory
 Chris Goddard. Advanced Topics in Information Dynamics. Has a chapter on Bott periodicity.
KTheory and real Bott periodicity
 Max Karoubi. Bott Periodicity in Topological, Algebraic and Hermitian KTheory
 Karoubi: Atiyah and Hirzebruch realized that Bott periodicity was related to the fundamental work of Grothendieck on algebraic Ktheory. By considering the category of topological vector bundles over a compact space {$X$} (instead of algebraic vector bundles), Atiyah and Hirzebruch defined a topological Ktheory {$K(X)$} following the same pattern as Grothendieck. As a new feature however, Atiyah and Hirzebruch managed to define “derived functors” {$K^{−n}(X)$} by considering vector bundles over the nth suspension of {$X_+$} ({$X$} with one point added outside). There are in fact two Ktheories involved, whether one considers real or complex vector bundles. We shall denote them by {$K_R$} and {$K_C$} respectively if we want to be specific. [Bott periodicity] is then equivalent to the periodicity of the functors {$K^{−n}$}. ... These isomorphisms enabled Atiyah and Hirzebruch to extend the definition of {$K^n$} for all {$n ∈ \mathbb{Z}$} and define what we now call a “generalized cohomology theory” on the category of compact spaces.
 Attiyah. Ktheory and Reality Quarterly Journal of Mathematics. Oxford. (2) , 17 (1966) 36786. Develops KR, a new Ktheory, that leads to an elegant proof of Bott periodicity for KOtheory.
 Clifford Modules by Atiyah, Bott, Schapiro, 1963 and their isomorphism of topological Ktheory and quotients of Clifford modules.
 Atiyah. Bott Periodicity and the Index of Elliptic Operators Proves both complex and real Bott periodicity. For the latter, discusses spinors, the decomposition of the total Spin bundle {$S=S^+\oplus S^$} and the total Dirac operator, which maps {$S^+$} to {$S^$} and vice versa, thus brings to mind the isometry {$J_3J_4$}.
 In the corresponding theory for the infinite orthogonal group, O, the space BO is the classifying space for stable real vector bundles. In this case, Bott periodicity states that, for the 8fold loop space...
Resources
 Guo Chuan Thiang. Lecture notes on symmetries, topological phases and Ktheory. A readable, helpful, concrete and broad overview of the math and physics, including the tenfold way of topological insulators, CPT symmetry, Clifford algebras, relating them to Ktheory.
 Domenico Monaco. Ktheory in condensed matter physics Course syllabus.
 Lecture 6 Handwritten notes: Grothendieck group of a semigroup. The K⁰group of a manifold. Reduced K⁰group and stable equivalence classes of vector bundles. Stable equivalence vs isomorphism; stable range condition. Topological constructions: wedge sum, smash product, reduced suspension. Computability of K⁰. Higher Kgroups. Bott periodicity. Application: the Ktheory of the ndimensional torus.
 References Husemoller and also Milnor, Stasheff. Characteristic Classes.
 Edward Witten. DBranes And KTheory. Dbrane charge takes values in the Ktheory of spacetime.
Alain Connes about Bott periodicity and CTP. "Why Four Dimensions and the Standard Model Coupled to Gravity...
Ideas
 Atiyah: Ktheory is related to quantum theory, cohomology is related to classical theory.
 Atiyah: Cohomology (graded by dimension, de Rham complex)(corresponds to classical theory) is related by the AH spectral sequence to Ktheory (and Clifford algebras) (corresponds to quantum theory)
 Laplace operator {$\Delta f = \nabla \cdot \nabla f$} is a differential operator given by the divergence ({$\nabla\cdot$}) of the gradient ({$\nabla f$}) of a scalar function {$f$} on Euclidean space.
 This suggests that the curl (between the divergence and the gradient) is trivial.
 Physically, the Laplacian describes diffusion.
 Laplace's equation is when the Laplacian is zero. Solutions are harmonic equations and are given by potential theory. They describe states of equilibrium, independent of time.
 The Laplacian is the simplest elliptic operator.
 {$Lu=\sum_{\alpha\leq m}a_\alpha(x)\partial^\alpha u$} where multiindex {$\alpha =(\alpha_1,\dots,\alpha_n)$} and the partial derivative has order {$\alpha_i$} in {$x_i$}.
 Elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highestorder derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions.
 Dirac operator is, conceptually, the square root of the Laplacian. {$D^2=\Delta$}
 Dirac operator is the basis for spinors and for Bott periodicity.
 Atiyah: Dirac operator is related to Bott periodicity and Ktheory.
 Dirac operator is the formal square root of the Laplace operator.
 The spinDirac operator (for a spin{$\frac{1}{2}$} particle confined in the plane) can be written in terms of the Pauli matrices {$D=i\sigma_x\partial_x  i\sigma_y\partial_y$}. What is this raised to the exponential?
Bottperiodicity also arises in string theory.
Ktheory and Neuroscience
Topological Analysis of Networks of Neurons
 Directed simplices. Streamling information.
 Directed flag complex.
 ndimensional cavities
 cohomology mod 2, homology nonzero
 if multiplication of classes is meaningful
 Steiner operations
 representations
 use tools of Hodge theory
Concepts to master in Ktheory and KOtheory
 Topological Ktheory
 EckmannHilton duality reverses the direction of all of the arrows in homotopy theory. An example is currying.
 Tensorhom adjunction Given rings {$R,S$}, fix an {$(R,S)$}bimodule {$X$}. Define functors {$F:\textrm{Mod}_R\rightarrow\textrm{Mod}_S$} and {$G:\textrm{Mod}_S\rightarrow\textrm{Mod}_R$} such that {$F(Y)=Y\otimes_R X$} for {$Y\in\textrm{Mod}_R$} and {$G(Z)=\textrm{Hom}_S(X,Z)$} for {$Z\in\textrm{Mod}_S$}. Then {$F$} is left adjoint to {$G$}: {$\textrm{Hom}_S(Y\otimes_R X,Z)\cong\textrm{Hom}_R(Y,\textrm{Hom}_S(X,Z))$}
 Currying, concretely, identifies maps {$X\times I\rightarrow Y$} with maps {$X\rightarrow Y^I$}.
 Reducedsuspension loopspace adjunction. Consider topological spaces and take {$I=[0,1]$}. The duality between {$X\times I$} and {$Y^I$} grounds the duality between the reduced suspension {$\Sigma X$} and the loop space {$\Omega Y$}. We have the adjunction {$(\Sigma X,Y)=(X,\Omega Y)$}. This allows the study of spectra, giving rise to cohomology.
 Reduced suspension takes a pointed space {$X$} with {$X_0$}, creates a cylinder {$X\times I$} and then equates all of the points on the axis {$x_0\times I$}, yielding {$\Sigma X$}.
 Loop space {$\Omega Y$} of a pointed topological space {$Y$} is the space of (based) loops in {$Y$}. This is equipped with the compactopen topology. A loop is a continuous pointed map from the pointed circle {$S^1$} to {$Y$}. The set of path components (equivalence classes) of {$\Omega Y$} is the fundamental group {$\pi_1(X)$}.
{$KO^{−i}(X)\equiv KO_0(Σ^iX)$}
 Classifying space {$BG$} of a topological group {$G$} is the quotient of a weakly contractible space {$EG$} (a topological space whose homotopy groups are trivial) by a proper free action of {$G$}
 For a discrete group {$G$}, {$BG$} is basically a pathconnected topological space {$X$} such that the fundamental group of {$X$} is {$G$} and the higher homotopy groups of {$X$} are trivial.
 Intuitively, I think it is the simplest space {$X$} whose fundamental group is {$G$}.
 Classifying space for U(n) BU can be thought of in two ways:
 the Grassmannian {$BU$} of nplanes in an infinitedimensional complex Hilbert space
 the direct limit, with the induced topology, of Grassmannians of n planes
 Classifying space for O(n) is the Grassmannian {$BO$} of nplanes in an infinitedimensional real space {$\mathbb {R}^{\infty}$}
 Puppe sequence is an iterated exact sequence based on a threecycle and the loop space. The coexact Puppe sequence, in the opposite direction, is an interated sequence based on the treecycle and the (reduced) suspension.
 For example, this long exact sequence is the homotopy sequence of a fibration.
{$$\dots\rightarrow \pi_nF\rightarrow \pi_nE\rightarrow \pi_nB\rightarrow \pi_{n1}F\rightarrow \dots$$}
 Fibration is a generalization of a fiber bundle.
 Fiber bundle is a space that is locally a product space {$B\times F$}, but globally in {$E$} may have a different topological structure
 Characteristic class associates to each principal bundle of X a cohomology class of X.
 Principal bundle is a fiber bundle {$\pi :P\rightarrow X$} with fiber {$G$} where {$G$} is the structure group and acts on the fiber by left multiplication.
 Clutching construction Via the clutching construction, homotopy groups of the stable space O are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: {$\pi_k(O)=\pi_{k+1}(BO)$}.
Suspension product
Smash product
Quasifibration
Related concepts
Noncommutative geometry
Infinite loop spaces. Loopspace suspension adjunction.
 J. F. Adams, Infinite Loop Spaces, Ann. of Math. Studies 90, 1978.
 4.66. If F→E→B is a fibration or fiber bundle with E contractible, then there is a weak homotopy equivalence F→ΩB .
 For each topological group G there is a fiber bundle G→EG→BG with EG contractible, hence by the proposition there is a weak equivalence G ≃ ΩBG. There is also a converse statement
 Combining this duality between Σ and Ω with the duality between fibers and cofibers, we see a duality relationship between the fibration and cofibration sequences of §4.3:
