nLab algebraic K-theory, a historical perspective

A little history: the beginnings

Algebraic K-theory grew out of two apparently unrelated areas of algebraic geometry and algebraic

topology. The second of these, historically, was the development by Grothendieck of (geometric and topological) K-theory based on projective modules over a ring, or finite dimensional vector bundles on a space, that is the Grothendieck group of the given ring or of the ring of functions on the space. (The connection between these is that the space of global sections of a finite dimensional vector bundle on a nice enough space, $X$ is a finitely generated projective module over the ring of continuous real or complex functions on $X$ . This latter aspect is where the link with topological K-theory comes in.)

The second input is from of simple homotopy theory. J. H. C. Whitehead, following on from earlier ideas of Reidemeister, looked at possible extensions of combinatorial group theory, with its study of presentations of groups, to give a combinatorial homotopy theory?. This would take the form of an ‘algebraic homotopy theory’ giving good algebraic models for homotopy types, and would hopefully ease the determination of homotopy equivalences for instance of polyhedra. The ‘combinatorial’ part was exemplified by his two papers on ‘Combinatorial Homotopy Theory’, but raised an interesting question.

Could one give a ‘combinatorial way of generating all homotopy equivalences, (up to homotopy), starting with some ‘elementary expansions’ and ‘contractions’?

He showed the answer was negative, and there was an invariant (Whitehead torsion) whose vanishing was a necessary and sufficient condition for a homotopy equivalence to be so constructible. That invariant was an element of a group constructed from the stable general linear group over the group ring of the fundamental group of the domain space. This was a quotient of what became known as $K_1(\mathbb{Z}\pi_1X)$, that is the abelianisation of $GL(\mathbb{Z}\pi_1X)$.

This ties in, even at this basic level with the nPOV and the processes around categorification. For a ring, $R$, the Grothendieck group, $K_0(R)$, looks at the core of the category of finitely generated projective modules, and takes its set of connected components, which is just the set of isomorphism classes. This becomes an abelian monoid under direct sum and then a group after group completion. For $K_1(R)$, you are taking the core again, but looking at the category of morphisms in it. If we take $K_0$ of that category we get $K_1$ of the ring. So $K_1$ looks at the loops whilst $K_0$ at the connected components.

References

One of the best accounts of the history of K-theory is by Chuck Weibel:

• Charles A. Weibel, The development of algebraic K-theory before 1980 published in Algebra, K-theory, groups, and education:

  on the occasion of Hyman Bass’s 65th birthday_, a volume of Contemporary Mathematics, American Mathematical Soc, 1999.

Other historical accounts include

• Dominique Arlettaz, Algebraic K-theory of rings from a topological viewpoint (pdf)

• Daniel Grayson, Quillen’s work in algebraic K-theory, J. K-Theory 11 (2013), 527–547 pdf