Algebraic K (3)

In the early days of algebraic K-theory, J.H.C. Whitehead introduced a group related to K1 for group rings. Henri Poincaré had attempted to define Betti numbers for a manifold through triangulation, but there was a gap in his proof that two triangulations of a manifold always yielded the same Betti numbers. Whitehead introduced the idea of simple homotopy type, which is defined by adding simplices to a complex in such a way that each additional simplex deformation retracts into a subdivision of the old space. He also introduced the torsion, an invariant that takes values in the Whitehead group and demonstrates that some homotopy equivalences are not simple. John Milnor later used Reidemeister torsion, a variant of Whitehead torsion, to disprove the conjecture that any two triangulations of a manifold admitted a common subdivision. Hyman Bass and Stephen Schanuel later provided the first adequate definition of K1 for a ring, and Bass' book Algebraic K-theory became a key reference for the subject. In the 1960s and 1970s, various definitions of higher K-theory were proposed, and Daniel Quillen's definition, which used exact categories and the Q-construction, became widely accepted. K-theory was seen as a homology theory for rings and a cohomology theory for varieties, but many of its theorems required regularity hypotheses. Applications of algebraic K-theory in topology included Whitehead torsion, Wall's finiteness obstruction, and the h- and s-cobordism theorems.

Algebraic K-theory was first defined by Hyman Bass and Stephen Schanuel, who used vector bundles on a suspension of a space to define K1 in topological K-theory. They also provided a definition of K0 for homomorphisms of rings, and showed that K0 and K1 could be fit together into a relative homology exact sequence. Bass' book "Algebraic K-theory" solidified and improved upon many known results in the field. Max Karoubi and Villamayor later defined well-behaved K-groups for all n, which are now called KVn and are related to homotopy-invariant modifications of K-theory. John Milnor defined K2 using the Steinberg group, and Hideya Matsumoto showed that for a field F, K2(F) is isomorphic to a certain quotient of F^x tensored with F^x. Daniel Quillen's definition of higher K-theory, using his "plus construction," and Segal's Γ-objects, which worked with isomorphisms of bundles, both made progress in understanding higher K-theory. Quillen's definition, in particular, was able to recover K1 and K2 and was widely accepted, but it did not give the correct value for K0 or negative K-groups. Quillen's definition also relied on the hypothesis that the ring or variety in question was regular. Later, Quillen's definition was improved upon by Waldhausen, who introduced Waldhausen categories and the simplicial category S⋅C. Waldhausen's algebraic K-theory of spaces, A-theory, played a similar role for higher K-groups as K1(Zπ1(M)) did for M, and he showed that there is a map from A(M) to a space Wh(M) whose homotopy fiber is a homology theory.

The study of higher K-theory in mathematics dates back to the late 1960s and early 1970s, when various definitions were proposed by Swan, Gersten, Nobile, and Villamayor, and Karoubi and Villamayor. These definitions allowed for the creation of K-groups, which are related to homotopy-invariant modifications of K-theory, and the computation of the K-groups of finite fields. However, the first definition to gain widespread acceptance was that of Daniel Quillen, who used an exact category and the Q-construction to build a category from short exact sequences. This definition allowed for the calculation of K0 and led to simpler proofs, but did not yield negative K-groups. Segal also introduced the concept of Γ-objects, which were derived from Grothendieck's definition of K0, but only applied to split exact sequences.

The early applications of algebraic K-theory in topology included the construction of Whitehead torsion and the study of h-cobordisms, which are manifolds that classify bundles of h-cobordisms. The s-cobordism theorem, which states that an h-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion vanishes, led to the introduction of the algebraic K-theory of spaces by Waldhausen. This theory provided a space, A(M), that played a similar role to K1(Zπ1(M)) for higher K-groups. In addition, the Brown-Gersten spectral sequence, which converges from the sheaf cohomology to the K-group of the total space, and Bloch's formula, were developed. Gersten also made the conjecture that Kn(R) injects into Kn(F) for a regular local ring R with fraction field F, but this has yet to be proven in general.

In the late 1960s and early 1970s, a number of definitions for higher K-theory were proposed, including those by Swan and Gersten, Nobile and Villamayor, and Karoubi and Villamayor. However, these definitions did not have all the expected properties. Later, it was discovered that Milnor K-theory is a direct summand of the true K-theory of a field. Daniel Quillen's definition, which related K-theory to the Adams operations, was widely accepted and led to the computation of K-groups for finite fields. However, Quillen's definition did not give the correct value for K0 or any negative K-groups. Segal's approach, called Γ-objects, worked with isomorphisms of bundles, but only applied to split exact sequences. Quillen's second definition, using exact categories, led to the construction of an auxiliary category using the Q-construction, which worked with short exact sequences and allowed for the computation of K0. This definition agreed with Quillen's first definition, but still did not yield negative K-groups. The localization exact sequence, a fundamental theorem in K-theory, was not known to hold in full generality, though it was known for regular rings and varieties and for the related theory of G-theory. Early work on higher K-theory often had regularity hypotheses. In topology, Whitehead torsion and h-cobordisms were related to K-theory. The s-cobordism theorem, which explains the relationship between h-cobordisms and Whitehead torsion, was placed in the context of the classifying space of h-cobordisms. Waldhausen introduced the algebraic

Algebraic K-theory is a branch of mathematics that focuses on the study of abelian categories, which are exact categories that have additional structure. This area of study was developed by Quillen, who used it to prove many fundamental theorems in algebraic K-theory. Additionally, Quillen was able to demonstrate that earlier definitions of Swan and Gersten were equivalent to his own under certain conditions. K-theory is known to be both a homology theory for rings and a cohomology theory for varieties, but early work in the field often relied on the assumption that the ring or variety in question was regular. Quillen was able to prove the existence of a localization exact sequence for a related theory called G-theory, which is defined as the K-theory of the category of coherent sheaves on a variety. However, he was unable to prove the existence of this sequence for K-theory in general.

One application of algebraic K-theory in topology is the construction of Whitehead torsion, which is related to the notion of h-cobordisms. H-cobordisms are manifolds that are the boundaries of larger manifolds, and the s-cobordism theorem states that an h-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion of one of the manifolds into the larger manifold vanishes. The classifying space of h-cobordisms, called A-theory, contains information about higher K-groups and is related to the Whitehead group of the fundamental group of a manifold.

Algebraic K-theory has also been connected to the study of singular varieties and étale cohomology. Quillen conjectured the existence of a spectral sequence relating étale cohomology to the l-adic completion of K-theory, and this was later generalized by Thomason using the derived category of sheaves. Other techniques for computing K-theory have been developed, including the Dennis trace map and the use of topological Hochschild homology and topological cyclic homology. These tools have allowed for further calculations in K-theory and have revealed connections to other areas of mathematics such as cyclic homology.

Algebraic K-theory is a branch of mathematics that studies the algebraic structures of rings and varieties. The lower K-groups, such as K0, are the first to be discovered and are defined as the Grothendieck group of finitely generated projective modules over a ring. Higher K-groups, like G-theory and K-theory, are defined in terms of categories of vector bundles or coherent sheaves. These theories have been used in the formulation of non-commutative main conjecture of Iwasawa theory and in the construction of higher regulators. There are also several important conjectures in algebraic K-theory, such as Parshin's conjecture which relates to the higher K-groups of smooth varieties over finite fields, and Bass' conjecture which states that all of the groups Gn(A) are finitely generated when A is a finitely generated Z-algebra.