Algebraic K (2)

John H.C. Whitehead introduced a group closely related to K1 for group rings, and Henri Poincaré attempted to define the Betti numbers of a manifold through triangulation. However, Poincaré's method had a flaw: he couldn't prove that two triangulations of a manifold always had the same Betti numbers. It was known that Betti numbers didn't change when the triangulation was divided, so it was clear that any two triangulations with a common division had the same Betti numbers. What wasn't known was if any two triangulations admitted a common division, a conjecture called the Hauptvermutung. Whitehead's realization that triangulations were stable under subdivision led him to create the concept of simple homotopy type. This type of equivalence is defined by adding simplices or cells to a simplicial complex or cell complex in a way that each new simplex or cell can be deformed back into a subdivision of the original space. Part of the motivation for this definition is that a triangulation's subdivision is simple homotopy equivalent to the original triangulation, so any two triangulations with a common subdivision must be simple homotopy equivalent. Whitehead proved that simple homotopy equivalence is a finer invariant than homotopy equivalence through the introduction of an invariant called torsion. Torsion, which takes values in a group now known as the Whitehead group and denoted as Wh(π), can distinguish between homotopy equivalences that are not simple. The Whitehead group was later found to be a quotient of K1(Zπ), where Zπ is the integral group ring of π. Later, John Milnor used Reidemeister torsion, an invariant related to Whitehead torsion, to disprove the Hauptvermutung.

Hyman Bass and Stephen Schanuel provided the first satisfactory definition of K1 for a ring. In topological K-theory, K1 is defined using vector bundles on a suspension of the space, all of which come from the clutching construction. This is a process where two trivial vector bundles on two halves of a space are joined along a common strip using gluing data from the general linear group. However, elements of that group that correspond to elementary matrices (matrices for elementary row or column operations) produce equivalent gluings. With this in mind, Bass and Schanuel defined K1 of a ring R as GL(R) / E(R), where GL(R) is the infinite general linear group (the union of all GLn(R)) and E(R) is the subgroup of elementary matrices. They also defined K0 of a homomorphism of rings and showed that K0 and K1 fit together in an exact sequence similar to the relative homology exact sequence.

The work on K-theory from this period was brought together in Bass's book, Algebraic K-theory. In addition to summarizing the known results, Bass improved the statements of many theorems. One notable example is the fundamental theorem of algebraic K-theory, which he proved with Murthy. This theorem describes K0 of a ring R in terms of K1 of R, the polynomial ring R[t], and the localization R[t, t−1] using a four-term exact sequence. Bass used this theorem to recursively define negative K-groups K−n(R). Max Karoubi also defined negative K-groups for certain categories and showed that his definitions resulted in the same groups as Bass's.

The next significant advancement in the field came with the definition of K2. Steinberg studied the universal central extensions of a Chevalley group over a field and provided a presentation of this group using generators and relations. In the case of the En(k) group of elementary matrices, the universal central extension is called the Steinberg group, Stn(k). In 1967, John Milnor defined K2(R) as the kernel of the homomorphism St(R) → E(R). K2 extended some of the known exact sequences for K1 and K0, and it had significant applications in number theory. For example, Hideya Matsumoto's 1968 thesis demonstrated that for a field F, K2(F) is isomorphic to F^×⊗ZF^×/<x⊗(1−x):x∈F∖{0,1}>. This relationship also holds for the Hilbert symbol, which determines the solvability of quadratic equations over local fields. Specifically, John Tate was able to prove that K2(Q) is closely related to the law of quadratic reciprocity.

In the late 1960s and early 1970s, various definitions of higher K-theory were proposed. Swan and Gersten both defined Kn for all n, and Gersten proved that his theory was equivalent to Swan's, although neither theory was known to possess all the desired properties. Nobile and Villamayor also proposed a definition of higher K-groups, and Karoubi and Villamayor defined well-behaved K-groups for all n. However, their equivalent of K1 was sometimes a proper quotient of the Bass-Schanuel K1. These K-groups are now called KVn and are related to homotopy-invariant modifications of K-theory. Inspired by Matsumoto's theorem, Milnor made a definition of higher K-groups for a field, which he referred to as "purely ad hoc." This definition didn't seem to generalize to all rings and wasn't known to be the correct definition of higher K-theory for fields. Later it was discovered by Nesterenko and Suslin and Totaro that Milnor K-theory is actually a direct summand of the true K-theory of the field. Specifically, K-groups have a filtration called the weight filtration, and the Milnor K-theory of a field is the highest weight-graded piece of the K-theory. Thomason also discovered that there is no analogue of Milnor K-theory for a general variety. The first widely accepted definition of higher K-theory was given by Daniel Quillen. As part of Quillen's work on the Adams conjecture in topology, he constructed maps from the classifying spaces BGL(Fq) to the homotopy fiber of ψq − 1, where ψq is the qth Adams operation acting on the classifying space BU. This map is acyclic, and after modifying BGL(Fq) slightly to produce a new space BGL(Fq)+, the map became a homotopy equivalence. This modification is known as the plus construction. Quillen realized that the Adams operations, which had previously been known to be related to Chern classes and K-theory, could be used to define the K-theory of R as the homotopy groups of BGL(R)+. This definition recovered K1 and K2 and allowed Quillen to compute the K-groups of finite fields.

Quillen's definition of K-theory failed to provide the correct value for K0 and didn't give any negative K-groups. The issue was that it was based on GL, which only considers gluing vector bundles and not the bundles themselves, so it couldn't describe K0. Segal introduced a new approach called Γ-objects, which used the bundles themselves and isomorphisms of the bundles as data to create a spectrum whose homotopy groups are the higher K-groups, including K0. However, Γ-objects could only impose relations for split exact sequences, not general exact sequences, so they could only be used to define the K-theory of a ring in the category of projective modules, where all short exact sequences split. In 1972, Quillen developed another approach to constructing higher K-theory that used an exact category, a category that satisfies similar but slightly weaker formal properties to the categories of modules or vector bundles. He constructed an auxiliary category using a device called the Q-construction, which builds a category using short exact sequences in the same way that Grothendieck's definition of K0 uses isomorphism classes of bundles. Quillen proved his "+ = Q theorem," showing that his two definitions of K-theory agreed with each other, which gave the correct K0 and led to simpler proofs, but still didn't provide any negative K-groups.

While all abelian categories are exact categories, not all exact categories are abelian. Quillen's use of exact categories allowed him to prove many of the basic theorems of algebraic K-theory and show that the earlier definitions of Swan and Gersten were equivalent to his under certain conditions. K-theory was understood to be a homology theory for rings and a cohomology theory for varieties, but many of its basic theorems required the ring or variety in question to be regular. Quillen was unable to prove the existence of the localization exact sequence, a fundamental relation, in full generality, but he was able to prove its existence for a related theory called G-theory. G-theory had been defined earlier by Grothendieck as the free abelian group on isomorphism classes of coherent sheaves on a variety, modulo relations coming from exact sequences of coherent sheaves. Quillen proved that for a regular ring or variety, K-theory equaled G-theory, so K-theory of regular varieties had a localization exact sequence. This led to the widespread use of regularity hypotheses in early work on higher K-theory.