Contents

<h2>Overview</h2>
<p>In algebraic topology, one tries to study topological spaces with the help of algebraic invariants such as the fundamental group or homology groups. (Topological) K-theory is also such an invariant (in fact, a generalized cohomology theory) with a very different flavor. The starting point of K-theory is the study of <em>vector bundles. </em>A vector bundle is a family of (real or complex) vector spaces parametrized by the points of a topological space. The idea of K-theory is to learn about a space by studying the equivalence classes of vector bundles parametrized by it. Vector bundles can be added (by taking the direct sum of vector spaces) and multiplied (by taking the tensor product), but they cannot be subtracted. One therefore defines the (zeroeth) K-group of a space <em>X</em> as the group associated to the monoid of vector bundles.</p>
<p>K-theory reveals many interesting features of a space that ordinary homology or homotopy cannot see. In addition, it has a strongly geometric appeal. For instance, we will use K-theory to prove the Hopf Invariant One Theorem:</p>
<p><strong>Theorem</strong> (Adams). The only normed division algebras over <strong>R</strong> are <strong>R</strong> itself, the complex numbers, the quaternions, and the octonions.</p>
<p>Further topics of the course are classifying spaces and characteristic classes.</p>
<h2>Essentials</h2>
<ul>
  <li><strong>Subject code</strong>: 405047</li>
  <li><strong>Time:</strong> Period 1+2 (week 36-50), Monday 13:00-15:45</li>
  <li><strong>Room:</strong> VU University Amsterdam, WN-S640</li>
  <li><strong>Credits:</strong> 6</li>
  <li><strong>Target groups:</strong> Master students</li>
  <li><strong>Prerequisites:</strong> A background in algebraic topology such as covered in a one-semester course (including fundamental groups, homology and cohomology groups)</li>
  <li><strong>Format:</strong> Lectures mixed with problem sessions, homework (not graded)</li>
  <li><strong>Examination:</strong> Either oral exam or a presentation in class, further: active participation in the problem session</li>
</ul>
<h2>Syllabus</h2>
<table width="100%" border="1" cellpadding="5">
  <tr>
    <th width="30%" scope="row">Sep 06, 2010</th>
    <td width="70%"><p>Introduction, definition of vector bundles, local and global trivializations, morphisms of vector bundles, definition of sections of a vector bundle, sections form a vector space and also a module over the ring of K-valued functions on the base space. Examples for vector bundles: tangent bundle of the n-sphere, Möbius bundle on the circle.</p>
      <p><em>Theorems</em>: 1) A morphism of vector bundles which induces a linear isomorphism on each fiber is an isomorphism. 2) Equivalent characterizations for a vector bundle to be trivial.</p>
    <p><em>Homework problems</em>: 1) Let K be  ℝ or ℂ and denote by KP<sup>n</sup> the real or complex projective n-space:</p>
    <p>KP<sup>n</sup> = {L &lt; K<sup>n+1</sup> | L one-dimensional K-subspace}</p>
    <p>Denote by E ➝ KP<sup>n</sup> the space E = {(L,v) | L ∈ KP<sup>n</sup>, v ∈ L} with its projection (L,v) ↦ L. Show that this is a one-dimensional vector bundle.</p>
    <p>2) Define a vector bundle on S<sup>n</sup> ⊂ ℝ<sup>n+1</sup> whose fiber at x are those vectors that are <em>normal</em> at x, i.e. are parallel to x. Is this bundle trivial?</p></td>
  </tr>
  <tr>
    <th scope="row">Sep 13, 2010</th>
    <td><p>Pullbacks of vector bundles. Direct sums of vector bundles (Whitney sum). Stable equivalence.<br />
      Example: the tangent bundle of the n-sphere is stably trivial. Example: the tangent bundle of real projective spaces.<br />
    Inner products on vector bundles.</p>
    <p><em>Theorems</em>: Any vector bundle on a paracompact base space has an inner product. Any subbundle of a vector bundle over a paracompact space has a complement.</p>
    <p><em>Homework problems:</em> 1) Show that any CW-complex is paracompact.</p>
    <p>2) Exercise 1.1.2 from Hatcher: for a subbundle E' ⊂ E ∈ Vect(B), construct a quotient bundle E/E', i.e. a vector bundle whose fiber at b ∈ B is E<sub>b</sub>/E'<sub>b</sub>.</p>
    <p>3) Give an example to show that for a map f: E ➝ E' in Vect(B), there may not exists a &quot;kernel&quot; vector bundle, i.e. a vector bundle whose fiber at b ∈ B is ker(E<sub>b</sub> ➝ E'<sub>b</sub>). What condition on f would be necessary and sufficient to ascertain the existence of a kernel?</p>
    <p>4) Let c: ℝP<sup>n</sup> ➝ ℂP<sup>n</sup> be the&quot;complexification&quot; map, which sends a point [x<sub>0</sub>,...,x<sub>n</sub>] of ℝP<sup>n</sup> to the point of the same name in ℂP<sup>n</sup>. Let L<sub>ℝ</sub> resp. L<sub>ℂ</sub> denote the tautological line bundles on ℝP<sup>n</sup> resp. ℂP<sup>n</sup>. Show that c*(L<sub>ℂ</sub>) ≅ L<sub>ℝ</sub> ⊕ L<sub>ℝ</sub>.</p></td>
  </tr>
  <tr>
    <th scope="row">Sep 20,2010</th>
    <td><div align="center"><em>No class</em></div></td>
  </tr>
  <tr>
    <th scope="row">Sep 29, 2010</th>
    <td><p><em>Theorem: </em>Any vector bundle over a compact space can be embedded into a trivial vector bundle.</p>
      <p>Definition of K(X), KO(X).</p>
    <p><em>Theorem: </em>If f, g: X ➝ Y are homotopic then f*E ≅ g*E for each vector bundle E over Y.</p>
    <p><a href="../../../pdf/topkthy/problems3.pdf">Homework problems</a></p>        </td>
  </tr>
  <tr>
    <th scope="row">Oct 04, 2010</th>
    <td><p>Construction of vector bundles with cocycles</p>
      <p>Clutching functions, Grassmann and Stiefel manifolds</p>
      <p><em>Theorem: </em>Vect<sup>n</sup>(X)/≅ = [X, G<sub>n</sub>] for compact X.</p>
      <p><a href="../../../pdf/topkthy/problems4.pdf">Homework problems</a></p></td>
  </tr>
  <tr>
    <th scope="row">Oct 11, 2010</th>
    <td><p>Cell decomposition of Grassmannians (Rik)</p>
    <p>Simplicial objects in a category, the simplicial set of singular simplices of a topological space, the geometric realization of a simplicial space.</p>
    <p><a href="../../../pdf/topkthy/problems5.pdf">Homework problems</a></p></td>
  </tr>
  <tr>
    <th scope="row">Oct 18, 2010</th>
    <td><p>Geometric realization commutes with products up to homeomorphism (Wicher)</p>
      <p>The nerve and the classifying space of a (topological) category, classifying spaces of groups. Begin of the proof of Bott periodicity according to Bruno Harris: <em>Bott periodicity via simplicial spaces</em>, J. Alg. <strong>62 </strong>(1980)</p></td>
  </tr>
  <tr>
    <th scope="row">Oct 25, 2010</th>
    <td>No lecture (exam period)</td>
  </tr>
  <tr>
    <th scope="row">Nov 1, 2010</th>
    <td>No lecture (sick)</td>
  </tr>
  <tr>
    <th scope="row">Nov 8, 2010</th>
    <td><p>Conclusion of the proof of Bott periodicity (modulo the group completion theorem and the fact that levelwise homotopy equivalences of simplicial spaces induce homotopy equivalences on the geometric realizations)</p>
    <p>Fiber bundles: definition, examples.</p>
    <p><a href="../../../pdf/topkthy/problems6.pdf">Homework problems</a></p></td>
  </tr>
  <tr>
    <th scope="row">Nov 15, 2010</th>
    <td><p>Fibrations. Definition and properties: fibrations are closed under composition, pullback. Mapping spaces of a given space into a fibration give fibrations, mapping spaces into a given space turn cofibrations into fibrations. Homotopy groups; the long exact sequence of homotopy groups for a fibration.</p>        </td>
  </tr>
  <tr>
    <th scope="row">Nov 22, 2010</th>
    <td><p>Shan: the group completion theorem.</p>
    <p>The Mayer-Vietoris sequence for homotopy groups of a pullback. The definition of a spectrum; a spectrum gives rise to a cohomology theory. Construction of K-theory as a spectrum.</p>
    <p><a href="../../../pdf/topkthy/problems7.pdf">Homework problems</a></p></td>
  </tr>
  <tr>
    <th scope="row">Nov 29, 2010</th>
    <td><p>The complex K-theory of CP<sup>n</sup>; the Leray-Hirsch theorem; projectivization of vector bundles; the splitting principle</p>
    <p><a href="../../../pdf/topkthy/problems8.pdf">Homework problems</a></p>        </td>
  </tr>
  <tr>
    <th scope="row">Dec 6, 2010</th>
    <td><p>Patrick: the Hurewicz theorem</p>
      <p>Adams operations and the Hopf invariant</p>
    </td>
  </tr>
  <tr>
    <th scope="row">Dec 13, 2010</th>
    <td><p>Robin: existence of Adams operations</p>
    <p>Hopf invariant 1, H-spaces, division algebras, parallelizability of spheres.</p></td>
  </tr>
</table>
<p>&nbsp;</p>
<h2>Literature</h2>
<ul>
  <li>Hatcher, A.: <a href="http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html"><em>Vector bundles and K-theory</em></a>. Book draft.</li>
  <li>Atiyah, M.: <em>K-theory</em>. Westview Press, 1994.</li>
  <li>Milnor, J. and Stasheff, J.: <em>Characteristic classes</em>. Princeton University Press, 1974.</li>
  <li>Husemoller, D.: <em>Fibre Bundles</em>, Springer GTM, 1994.</li>
</ul>