Andrius Kulikauskas

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Introduction E9F5FC

Questions FFFFC0


Investigation: Understand K-theory and KO-theory as relevant for Bott periodicity.


  • 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?

K-theory group {$K(X)$}

  • The K-theory 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 K-theory.


K-theory definition
  • The study of a ring generated by vector bundles over a topological space or scheme.
  • Involves the construction of families of K-functors 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

K-theory goals

  • How dimensions are related
  • Cohomology - sequence of groups - which may not exist. Motivic cohomology
  • K-theory was used for the first proof of the Atiyah-Singer index theorem. They reduced the index theorem to the case of spheres, and then to the case of a point.

K-theory ideas

  • When you move from K-theory to KO-theory, 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.



K-theory and complex Bott periodicity

K-Theory and real Bott periodicity

  • Max Karoubi. Bott Periodicity in Topological, Algebraic and Hermitian K-Theory
    • Karoubi: Atiyah and Hirzebruch realized that Bott periodicity was related to the fundamental work of Grothendieck on algebraic K-theory. By considering the category of topological vector bundles over a compact space {$X$} (instead of algebraic vector bundles), Atiyah and Hirzebruch defined a topological K-theory {$K(X)$} 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 K-theories 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. K-theory and Reality Quarterly Journal of Mathematics. Oxford. (2) , 17 (1966) 367-86. Develops KR, a new K-theory, that leads to an elegant proof of Bott periodicity for KO-theory.
  • Clifford Modules by Atiyah, Bott, Schapiro, 1963 and their isomorphism of topological K-theory 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 8-fold loop space...

K-theory and physics


Alain Connes about Bott periodicity and CTP. "Why Four Dimensions and the Standard Model Coupled to Gravity...


  • Atiyah: K-theory 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 K-theory (and Clifford algebras) (corresponds to quantum theory)
Laplace operator or Laplacian
  • 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 multi-index {$\alpha =(\alpha_1,\dots,\alpha_n)$} 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 highest-order 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 K-theory.
  • Dirac operator is the formal square root of the Laplace operator.
  • The spin-Dirac 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?

Bott-periodicity also arises in string theory.

K-theory and Neuroscience

Topological Analysis of Networks of Neurons

  • Directed simplices. Streamling information.
  • Directed flag complex.
  • n-dimensional cavities
  • cohomology mod 2, homology nonzero
  • if multiplication of classes is meaningful
  • Steiner operations
  • representations
  • use tools of Hodge theory

Concepts to master in K-theory and KO-theory

  • Topological K-theory
  • Eckmann-Hilton duality reverses the direction of all of the arrows in homotopy theory. An example is currying.
    • Tensor-hom 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 {$X\rightarrow Y^I$}.
    • Reduced-suspension 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 compact-open 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 path-connected 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 n-planes in an infinite-dimensional 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 n-planes in an infinite-dimensional real space {$\mathbb {R}^{\infty}$}
  • Puppe sequence is an iterated exact sequence based on a three-cycle and the loop space. The coexact Puppe sequence, in the opposite direction, is an interated sequence based on the tree-cycle and the (reduced) suspension.
    • For example, this long exact sequence is the homotopy sequence of a fibration.

{$$\dots\rightarrow \pi_nF\rightarrow \pi_nE\rightarrow \pi_nB\rightarrow \pi_{n-1}F\rightarrow \dots$$}

  • 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


Related concepts

Extended topics

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:
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