Editing Michael Atiyah (section)

===K-theory (1959–1974)===
{{Main|K-theory}}
[[File:Möbius strip.jpg|thumb|right|250px|A [[Möbius band]] is the simplest non-trivial example of a [[vector bundle]].]]
Atiyah's works on [[K-theory]], including his book on K-theory<ref>{{harvnb|Atiyah|1989}}</ref> are reprinted in volume 2 of his collected works.<ref>{{harvnb|Atiyah|1988b}}</ref>

The simplest nontrivial example of a vector bundle is the [[Möbius band]] (pictured on the right): a strip of paper with a twist in it, which represents a rank 1 vector bundle over a circle (the circle in question being the centerline of the Möbius band). K-theory is a tool for working with higher-dimensional analogues of this example, or in other words for describing higher-dimensional twistings: elements of the K-group of a space are represented by vector bundles over it, so the Möbius band represents an element of the K-group of a circle.<ref>{{cite arXiv |last1=Atiyah |first1=Michael |author1-link=Michael Atiyah |year=2000 |title=K-Theory Past and Present |eprint=math/0012213}}</ref>

Topological K-theory was discovered by Atiyah and [[Friedrich Hirzebruch]]<ref>{{harvnb|Atiyah|1988b|loc=paper 24}}</ref> who were inspired by Grothendieck's proof of the [[Grothendieck–Riemann–Roch theorem]] and Bott's work on the [[Bott periodicity theorem|periodicity theorem]]. This paper only discussed the zeroth K-group; they shortly after extended it to K-groups of all degrees,<ref name="paper28">{{harvnb|Atiyah|1988b|loc=paper 28}}</ref> giving the first (nontrivial) example of a [[generalized cohomology theory]].

Several results showed that the newly introduced K-theory was in some ways more powerful than ordinary cohomology theory. Atiyah and Todd<ref>{{harvnb|Atiyah|1988b|loc=paper 26}}</ref> used K-theory to improve the lower bounds found using ordinary cohomology by Borel and Serre for the [[James number]], describing when a map from a complex [[Stiefel manifold]] to a sphere has a cross section. ([[J. Frank Adams|Adams]] and Grant-Walker later showed that the bound found by Atiyah and Todd was best possible.) Atiyah and Hirzebruch<ref>{{harvnb|Atiyah|1988a|loc=papers 30,31}}</ref> used K-theory to explain some relations between [[Steenrod operation]]s and [[Todd class]]es that Hirzebruch had noticed a few years before. The original solution of the [[Hopf invariant one problem]] operations by J. F. Adams was very long and complicated, using secondary cohomology operations. Atiyah showed how primary operations in K-theory could be used to give a short solution taking only a few lines, and in joint work with Adams<ref>{{harvnb|Atiyah|1988b|loc=paper 42}}</ref> also proved analogues of the result at odd primes.

[[File:Atiyah-Hirzebruch.jpeg|thumb|right|250px|Michael Atiyah and [[Friedrich Hirzebruch]] (right), the creators of K-theory]]
The [[Atiyah–Hirzebruch spectral sequence]] relates the ordinary cohomology of a space to its generalized cohomology theory.<ref name="paper28" /> (Atiyah and Hirzebruch used the case of K-theory, but their method works for all cohomology theories).

Atiyah showed<ref>{{harvnb|Atiyah|1961}}</ref> that for a finite group ''G'', the K theory of its [[classifying space]], ''BG'', is isomorphic to the [[completion (ring theory)|completion]] of its [[representation ring|character ring]]:
:<math> K(BG) \cong R(G)^{\wedge}.</math>
The same year<ref>{{harvnb|Atiyah|Hirzebruch|1961}}</ref> they proved the result for ''G'' any [[Compact group|compact]] [[Connected space|connected]] [[Lie group]]. Although soon the result could be extended to ''all'' compact Lie groups by incorporating results from [[Graeme Segal]]'s thesis,<ref>{{harvnb|Segal|1968}}</ref> that extension was complicated. However a simpler and more general proof was produced by introducing [[Equivariant algebraic K-theory|equivariant K-theory]], ''i.e.'' equivalence classes of ''G''-vector bundles over a compact ''G''-space ''X''.<ref>{{harvnb|Atiyah|Segal|1969}}</ref> It was shown that under suitable conditions the completion of the equivariant K theory of ''X'' is [[isomorphic]] to the ordinary K-theory of a space, <math>X_G</math>, which fibred over ''BG'' with fibre ''X'':
:<math>K_G(X)^{\wedge} \cong K(X_G). </math>

The original result then followed as a corollary by taking ''X'' to be a point: the left hand side reduced to the completion of ''R(G)'' and the right to ''K(BG)''. See [[Atiyah–Segal completion theorem]] for more details.

He defined new generalized homology and cohomology theories called bordism and [[cobordism]], and pointed out that many of the deep results on cobordism of manifolds found by [[René Thom]], [[C. T. C. Wall]], and others could be naturally reinterpreted as statements about these cohomology theories.<ref>{{harvnb|Atiyah|1988b|loc=paper 34}}</ref> Some of these cohomology theories, in particular complex cobordism, turned out to be some of the most powerful cohomology theories known.

{{quote box
| align = right
| width = 33%
| quote = "Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine."
| source = Michael Atiyah<ref>{{harvnb|Atiyah|2004|loc = paper 160, p. 7}}</ref>
}}
He introduced<ref name="paper37">{{harvnb|Atiyah|1988b|loc=paper 37}}</ref> the [[J-group]] ''J''(''X'') of a finite complex ''X'', defined as the group of stable fiber homotopy equivalence classes of [[sphere bundle]]s; this was later studied in detail by [[J. F. Adams]] in a series of papers, leading to the [[Adams conjecture]].

With Hirzebruch he extended the [[Grothendieck–Riemann–Roch theorem]] to complex analytic embeddings,<ref name="paper37" /> and in a related paper<ref>{{harvnb|Atiyah|1988b|loc=paper 36}}</ref> they showed that the [[Hodge conjecture]] for integral cohomology is false. The Hodge conjecture for rational cohomology is, as of 2008, a major unsolved problem.<ref>{{citation|url=http://www.claymath.org/millennium/Hodge_Conjecture/Official_Problem_Description.pdf|publisher=The Clay Math Institute|title=The Hodge conjecture|first=Pierre|last=Deligne|access-date=14 August 2008|archive-url=https://web.archive.org/web/20080827172255/http://www.claymath.org/millennium/Hodge_Conjecture/Official_Problem_Description.pdf|archive-date=27 August 2008}}</ref>

The [[Bott periodicity theorem]] was a central theme in Atiyah's work on K-theory, and he repeatedly returned to it, reworking the proof several times to understand it better. With Bott he worked out an elementary proof,<ref>{{harvnb|Atiyah|1988b|loc=paper 40}}</ref> and gave another version of it in his book.<ref>{{harvnb|Atiyah|1988b|loc=paper 45}}</ref> With Bott and [[Alexander Shapiro|Shapiro]] he analysed the relation of Bott periodicity to the periodicity of [[Clifford algebras]];<ref>{{harvnb|Atiyah|1988b|loc=paper 39}}</ref> although this paper did not have a proof of the periodicity theorem, a proof along similar lines was shortly afterwards found by R. Wood.  He found a proof of several generalizations using [[elliptic operator]]s;<ref>{{harvnb|Atiyah|1988b|loc=paper 46}}</ref> this new proof used an idea that he used to give a particularly short and easy proof of Bott's original periodicity theorem.<ref>{{harvnb|Atiyah|1988b|loc=paper 48}}</ref>