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Home , Bott periodicity theorem

Attitudes of 퐾-theory Topological, Algebraic, Combinatorial

Inna Zakharevich

1034 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7 Introduction: Divide and Conquer to study a (compact Hausdorff) space by examining the In many areas of we use similar methods for ways that vector bundles on that space behave. A vector analyzing problems. One of the most common is known bundle on a space 푋 is a continuous family of vector spaces as “divide and conquer”: split up a problem into smaller indexed by 푋. Somewhat more precisely, it is a space 퐸 to- problems, solve each of the smaller problems, then glue gether with a map 푝 ∶ 퐸 푋, such that for any 푥 ∈ −1 the solutions back together into a solution to the whole 푋, 푝 (푥) is a vector space (and all of these fit together −1 thing. Often, at the end, there is a frustrating question left nicely). We call 푝 (푥) the fiber over 푥, and denote it 퐸푥. over: These 퐸푥 are exactly the vector spaces in our continuous Is the constructed solution the only one possible? family. In other words, do the parts of the solution We can also think of vector bundles more locally. An 푋 uniquely determine the solution to the whole? example of a on is the “trivial bundle” 푋 × 퐑푛. A general vector bundle looks like a trivial bun- One approach to solving this question is that of consider- dle in a neighborhood of any point. Thus we can assem- ing only locally defined objects. For example, if two func- ble a bundle by taking several of these and gluing them tions agree at all points then they are equal; thus the an- together. More precisely, we can cover 푋 by a family of swer to the question is always “yes.” Sometimes, however, open subsets {푈훼}훼∈퐴, and think of the vector bundle as this is not possible, and it is necessary to analyze all of the 푛 looking like 푈훼 × 퐑 over each 푈훼. We then “glue to- different ways that pieces can be put together. For exam- gether” these bundles into a bundle on 푋 similarly to how ple, even if it is known that an abelian group has a filtra- manifolds are glued together: by giving gluing data that 퐙/2 ⊕ 퐙/2 tion with associated graded , there are multiple tell us how to identify the part of 푈 × 퐑푛 that sits over 퐙/2 ⊕ 퐙/2 훼 possibilities for what the group could be: or 푈 ∩ 푈 to the part of 푈 × 퐑푛 that sits over 푈 ∩ 푈 . 퐙/4 퐾 훼 훽 훽 훼 훽 . This is where -theory comes in. One requires that this gluing data should restrict to a lin- 퐾-theory is the study of invariants of assembly prob- 푛 ear isomorphism from the copy of 퐑 over 푥 in 푈훼 to the lems. Topological 퐾-theory studies how vector bundles are 푛 copy of 퐑 over 푥 in 푈훽; this gives an element of GL푛(퐑) assembled; algebraic 퐾-theory studies how modules over for every 푥 ∈ 푈훼 ∩ 푈훽. This gluing data should be con- a ring are assembled; other kinds of 퐾-theory study how tinuous; this means that we want continuous maps 푔훼훽 ∶ other kinds of objects can be assembled. The general ap- 푈훼 ∩푈훽 GL푛(퐑) for all 훼, 훽. Assuming that these sat- proach is to define a group 퐾0 with a generator for each isfy some standard relations (for example, that 푔훼훽(푥) = object of study and a relation for each possible “assem- 푔 (푥)−1 for all 푥 ∈ 푈 ∩ 푈 ) this gives enough data to bly.” Because different areas of mathematics have different 훽훼 훼 훽 construct the vector bundle 퐸. (For a more precise treat- tools and approaches, the question of classifying different ment, see for example [11, Chapter 1].) forms of 퐾-theory varies drastically in difficulty from field Two vector bundles 푝 ∶ 퐸 푋 and 푝′ ∶ 퐸′ 푋 to field. are considered isomorphic if there is a homeomorphism In this article we give a brief overview of three differ- 푓 ∶ 퐸 퐸′ such that for all 푥 ∈ 푋, the restriction of 푓 ent kinds of 퐾-theories: topological, algebraic, and com- to 퐸 gives a linear isomorphism to 퐸′ . In other words, 푓 binatorial. Between the sections we give some context and 푥 푥 must satisfy 푝′ ∘ 푓 = 푝. background to explain how the modern perspective on 퐾- 푋 = 푆1 theory developed, and some of the motivation for the the- Let us consider an example. Suppose that and 푛 = 1 푆1 ory behind it. Although this article is not (and cannot) be . To each point in we attach a line, in such a a comprehensive history of or motivation for the topic, we way that these lines form a continuous family; thus we can hope that it will be an interesting perspective on the devel- think of this as attaching an infinitely wide strip of paper to opment of this field. a circle. Starting at a point in the circle, let us walk around clockwise and try to make sure that the paper is standing Topological 퐾-Theory up “vertically” (compared to the plane of the circle). When The first example of 퐾-theory that most people see is topo- we get back to the beginning one of two things can happen: logical 퐾-theory. The idea behind topological 퐾-theory is either the entire strip is standing up vertically (in which case we have the bundle 푆1 × 퐑1) or we failed: there is a Inna Zakharevich is an assistant professor of mathematics at Cornell University. twist in the paper, so that we have a Möbius strip. These are Her email address is [email protected]. the two possibilities for one-dimensional vector bundles The author was supported in part by NSF grant DMS-1654522. on a circle. Communicated by Notices Associate Editor Daniel Krashen. A vector bundle is fundamentally a geometric object, For permission to reprint this article, please contact: but it turns out to be classified by homotopical information. [email protected]. For a concrete example, let us consider 푋 = 푆푛 and cover 푛 푛 DOI: https://doi.org/10.1090/noti1908 it by the family {푆 ∖{north pole}, 푆 ∖{south pole}}.

AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1035 An 푛-dimensional vector bundle on 푋 is then uniquely de- We can then define addition by 푛 termined by a single function 푆 ∖ {poles} GL푛(퐑). (퐸, 퐸′) + (퐹, 퐹′) = (퐸 ⊕ 퐹, 퐸′ ⊕ 퐹′) It turns out that the isomorphism class of the vector bun- dle depends only on the class of this map, not and multiplication by on the actual map; in particular, we can consider it to be (퐸, 퐸′)(퐹, 퐹′) = ((퐸⊗퐹)⊕(퐸′⊗퐹′), (퐸⊗퐹′)⊕(퐸′⊗퐹). 푛−1 a map from 푆 GL푛(퐑) (see [11, Section 1.2]). For example, applying this to the circle case above we see that These operations define a ring structure on 0 def we have four homotopy classes of maps 푆 = {±1} 퐾푂0(푋) = Vect(푋)2/ ∼ . GL1(퐑), depending only on the sign of the image of 0 each of the two points. By changing the orientation on (The group 퐾 (푋) is defined analogously using complex one of the two patches, we can assume that the point 1 vector bundles.) 0 is mapped to the identity transformation, which gives us It turns out [11, Chapter 2] that 퐾푂 (푋) is a homotopy two possibilities: if the image of −1 is negative in GL1(퐑) invariant. Moreover, if we define then we get a Möbius strip, and otherwise we get the triv- def 퐾푂−푛(푋) = 퐾0(Σ푛(푋 )) 푛 ≥ 0 ial bundle. This confirms our above intuitive description + 1 푛 of the possible bundles on 푆 . (where Σ (푋+) is the reduced of 푋 with a dis- With this perspective we see that the geometric data of joint basepoint added [12, Example 0.10]) it turns out that the vector bundle is controlled by the homotopical data of these groups have an 8-fold periodicity. (This is called the the gluing maps. In fact, this kind of construction works Bott periodicity theorem [5, 1.15].) This allows us to extend more generally. Let Vect푛(푋) be the set of isomorphism this definition to all integers 푛. With these definitions ∗ classes of vector bundles on 푋. 퐾푂 (푋) has been used to great advantage to solve var- ious geometric problems. The two most famous are the Theorem 1 ([11, Theorem 1.16]). There exists a space called following: 퐵푂(푛) such that for finite CW complexes1 푋, Hopf invariant 1: There are some standard examples of 푛 Vect (푋) = [푋, 퐵푂(푛)] spheres that have unital multiplications, starting with 푆0 푆1 where the right-hand side denotes the homotopy classes of maps and , which both have abelian group structures. 푆0 1 퐑 푋 퐵푂(푛). By noting that is the elements of norm in and 푆1 is the elements of norm 1 in 퐂 we can construct One way of understanding this theorem is to say, as we similar multiplications on 푆3 and 푆7, viewing them did above, that the geometric data of vector bundles is con- as the units in the and octonions. It turns trolled by homotopical data; another way is to say that the out that these are the only examples of spheres with 푛 geometric data of Vect (푋) contains homotopical infor- unital multiplication; this was originally proved by mation about 푋. Adams in [1] using the Steenrod algebra, but a much We can develop this idea into a very powerful theory simpler and cleaner proof was discovered by Adams called topological 퐾-theory. and Atiyah [3] using topological 퐾-theory. Let us consider the set of isomorphism classes of vector This problem is closely related to other classical bundles on 푋, which we write Vect(푋). This set has two questions, including the existence of normed division operations on it: ⊕ and ⊗. For any two vector bundles algebra structures on 퐑푛, the parallelizability of 푆푛−1, ′ ′ 퐸, 퐸 over 푋 we can construct a vector bundle 퐸⊕퐸 where and the existence of spaces with (graded commuta- ′ ′ ′ we set (퐸 ⊕ 퐸 )푥 = 퐸푥 ⊕ 퐸푥, or a vector bundle 퐸 ⊗ 퐸 tive) polynomial . ′ ′ by (퐸 ⊗ 퐸 )푥 = 퐸푥 ⊗ 퐸푥. These two operations are unital: Counting vector fields on spheres: The hairy ball theo- 0 the 0-dimensional bundle 푋 × 퐑 is the unit for ⊕, and rem states that there is no nonvanishing continuous 1 the 1-dimensional bundle 푋 × 퐑 is the unit for ⊗. Thus vector field on 푆2. However, for odd-dimensional the only thing stopping us from having a ring of vector spheres such fields exist, and so the more complicated bundles on 푋 is that we have no additive inverses. question becomes: what is the maximum number of The solution to this is to add them formally, the same everywhere-linearly-independent vector fields that can way as we build the rational numbers from the integers. be put on a sphere? Adams [2] showed, using topo- ′ 2 Consider a pair (퐸, 퐸 ) ∈ Vect(푋) , which we think of logical 퐾-theory, that there are 휌(푛) − 1 such fields ′ as a “formal difference” 퐸 − 퐸 . We define an equivalence on 푆푛−1, where 휌(푛) is the Radon–Hurwitz number. ′ ′ ′ ′ relation (퐸, 퐸 ) ∼ (퐹, 퐹 ) if 퐸⊕퐹 ≅ 퐸 ⊕퐹, exactly anal- (Prior to Adams it was known that there are at least 푎 푐 ogously to how two fractions 푏 and 푑 are equal if 푎푑 = 푏푐. that many fields; using the structure of topological 퐾- theory Adams was able to show that this is a sharp 1Actually, all that is required is that 푋 is paracompact. bound.)

1036 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7 It turns out that topological 퐾-theory is a generalized co- In fact, the Brown Representability Theorem [6] states theory: a way of assigning groups to spaces that that such a construction is possible for any generalized co- behaves just like singular cohomology, but which does not homology theory. It thus makes sense to think of a co- satisfy the dimension axiom.2 Thus all of the standard homology theory not as a sequence of functors, but in- computational tools that work for ordinary cohomology stead as a sequence of spaces. The only extra data nec- theory work for topological 퐾-theory. For example, for a essary to produce a cohomology theory from a sequence sub-CW-complex 푌 of a finite CW-complex 푋 there exists of spaces is structure maps, which are weak equivalences ∼ a long exact sequence 푋푛 ⟶ Ω푋푛+1. Given this extra data, it makes sense to ⋯ 퐾푂푛(푋/푌) 퐾푂푛(푋) 퐾푂푛(푌) drop all spaces with negative indices and assume that 푛+1 푛 퐾푂 (푋/푌) ⋯ . 푋−푛 ≃ Ω 푋0 for 푛 > 0.

These new invariants, together with their geometric under- A space 푋0 that can appear at the beginning of such a se- pinnings, are what give topological 퐾-theory its power. quence is called an infinite-, since it is weakly equivalent to an 푛-th loop space for all 푛. In this case, Segue: Representable Cohomology Theories the groups 휋푛푋0 are exactly the cohomology groups of The group 퐾푂0(푋) was defined algebraically in the pre- the sphere in the cohomology theory represented by this vious section, but it is in fact possible to define it com- sequence. pletely homotopically. If we write [푋, 푌] for the set of It is common, therefore, to shift perspective from con- (basepointed) homotopy classes of maps from 푋 to 푌, it structing a cohomology theory to constructing a space turns out that (which turns out to be an infinite-loop space) whose ho- 퐾푂0(푋) ≅ [푋 , 퐙 × 퐵푂], motopy groups are exactly the cohomology groups of the + sphere (for the cohomology theory in question). Such a where 푋+ is 푋 with a disjoint basepoint added, and 퐵푂 is space contains all of the information of the cohomology the delooping3 of the infinite (in which theory, but has much more structure to work with. elements are orthogonal matrices that differ from the iden- tity in only a finite number of spaces) [11, Section 1.2]. Algebraic 퐾-Theory The space 퐵푂 is closely related to the spaces 퐵푂(푛) men- The next step is to construct a version of 퐾-theory for rings, tioned above; the difference is that previously we were con- rather than spaces. It is important to note that topologi- sidering vector bundles of a fixed dimension, while here cal and algebraic 퐾-theory are completely distinct, both in we consider all vector bundles at once. Using the above approach and results: topological 퐾-theory starts with a expression we can then see that space and constructs a ring, while algebraic 퐾-theory starts with a ring and constructs a space. The connection be- 퐾푂−푛(푋) ≅ [Σ푛(푋 ), 퐙 × 퐵푂] ≅ [푋 ,Ω푛(퐙 × 퐵푂)], + + tween them is the spirit of the Serre–Swan theorem: where Ω푌 is the loop space of 푌, whose points are (base- Theorem 2 (Serre–Swan). The category of real vector bun- pointed) maps 푆1 푌. Bott periodicity simply states dles over a compact 푋 is equivalent to the category of finitely that 퐶(푋) 8 generated projective modules over the ring of continuous Ω (퐙 × 퐵푂) ≃ 퐙 × 퐵푂. real-valued functions on 푋. Using these results we can construct a sequence of spaces This theorem is motivation for starting with the follow- … , 퐾−2, 퐾−1, 퐾0, 퐾1, 퐾2,… ing analogy: vector bundles over a space are analogous to 푛 finitely generated projective modules over a ring. such that 퐾푂 (푋) ≅ [푋+, 퐾푛]. 퐾푂0(푋) A similar statement can be made for ordinary cohomol- We can rewrite the definition of to be the free abelian group generated by vector bundles over 푋, mod- ogy theory. Letting 퐸푛 = 퐾(퐙, 푛) be the 푛-th Eilenberg– ulo the relation that [퐸 ⊕ 퐸′] = [퐸] + [퐸′]. Analo- MacLane space of 퐙 [12, Section 4.2], and setting 퐸푛 = ∗ 퐾 (푅) for 푛 < 0 we have gously, we define the group 0 to be the group gen- erated by isomorphism classes of finitely generated pro- 푛 퐻 (푋; 퐙) ≅ [푋+, 퐸푛]. jective 푅-modules, modulo the relation that [퐴 ⊕ 퐵] = [퐴] + [퐵]. At this point, when working with topological 2The dimension axiom states that all cohomology groups of a point other than the 0-th one 퐾-theory, we were done: we could use suspension to de- must be 0. This is what allows us to read off the “dimension” of a space from its cohomol- ogy. Topological 퐾-theory doesn’t work this way: the topological 퐾-theory of a point has fine the negative 퐾-groups, and Bott periodicity handled infinitely many nonzero groups. the rest. Unfortunately, an analogous definition generally 3The delooping of a space 푌 is a space 푍 such that 푌 ≃ Ω푍. The delooping of a discrete group is just the of the group; however, since 푂 has a nondiscrete topology produces no new useful groups, and no Bott periodicity; a bit more care is necessary. thus a different approach is needed in this situation.

AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1037 The Serre–Swan theorem also hints that we must change 푝(퐴 ⊕ 퐼) = 푝(퐴) = 푝(퐼 ⊕ 퐴). Thus we must have our grading, from cohomological to homological. Given a 푝(퐴)푝(퐵) = 푝(퐴 ⊕ 퐼)푝(퐼 ⊕ 퐵) = 푝((퐴 ⊕ 퐼)(퐼 ⊕ 퐵)) map 푌 푋 of spaces, the induced map on rings of func- tions goes in the opposite direction: 퐶(푋) 퐶(푌). The = 푝(diag(퐴, 퐵)) same will hold for our 퐾-theories: previously, if we had a and analogously 푝(퐴)푝(퐵) = 푝(diag(퐵, 퐴). But by this 0 0 map 푌 푋 it induced a map 퐾 (푋) 퐾 (푌) by same argument, 푝(퐵)푝(퐴) = 푝(diag(퐴, 퐵)), so 퐾1(푅) pulling back vector bundles. Now, on the other hand, we must be abelian. It turns out that defining have the opposite map: given a homomorphism 푆 푅 def ab of rings we get a map 퐾0(푆) 퐾0(푅) by tensoring an 퐾1(푅) = 퐺퐿(푅) 푆-module up to an 푅-module. In addition, this switch re- produces a stable invariant. When 푅 is a field this is just verses the signs of the 퐾-groups, with 퐾1(푅) playing an the group of invertible elements in 푅, and the map 퐺퐿(푅) −1 analogous role to 퐾푂 (푋). 퐾1(푅) takes the determinant of a matrix. Topological 퐾-theory is a generalized cohomology theory, whose defining property is the existence of along Example. To see that this is a reasonable definition, at exact sequence. We are going to keep this, as well as the least for fields, let us consider the example when 푅 is a Dedekind domain with fraction field 퐹. In this case [22, definition of 퐾0. The imagined leap of faith is the obser- vation that long exact sequences appear naturally for the Corollary II.2.6.3] we have 퐾0(푅) ≅ 퐙 ⊕ Cl(푅), where homotopy groups of topological spaces. This is what we Cl(푅) is the class group of 푅. Since 퐾0(퐹) ≅ 퐙, the defi- are going to construct: given 푅, we want to have a topo- nition of Cl(푅) leads to an exact sequence def × × logical space 퐊(푅) and define 퐾푖(푅) = 휋푖퐊(푅). This 푅 퐹 ⨁ 퐙 퐾0(푅) 퐾0(퐹) 픭 prime space should have the property that 휋0퐊(푅) ≅ 퐾0(푅), and such that its higher homotopy groups are meaning- 0. ful algebraic invariants; in particular, at 퐾1(푅) and 퐾2(푅) Note that each term in the infinite sum is actually 퐾0(푅/픭). they should agree with classically-defined invariants (for The determinant gives a surjective map 퐺퐿(푅) 푅×, so more on this, see [22, Chapters II, III]). × in fact there is a surjective map 퐾1(푅) 푅 . We can 퐾 (푅) 푅 One can see that 0 encodes -modules up to cer- therefore rewrite the above sequence as tain transformations: isomorphism and stability. Thus 휋1퐊(푅) should encode loops of such transformations. Let 퐾1(푅) 퐾1(퐹) ⨁ 퐾0(푅/픭) 퐾0(푅) us consider isomorphisms first. If we think of a path be- 픭 prime tween modules as an isomorphism between them, then a 퐾0(퐹) 0. loop of such transformations is simply an automorphism This may be considered another confirmation of the cor- of a module. To make this into a group we need to think of 4 rectness of our definition of 퐾1(푅). the automorphisms of 푅-modules as all living in the same place; fortunately, all finitely generated projective modules We now turn to a construction of 퐊(푅). A naive way to can be thought of as summands of the same module, 푅∞. begin is to try and construct a CW complex whose points An automorphism of a module 푀 sitting inside 푀⊕푁 can are represented by finitely generated projective 푅-modules, be thought of as an automorphism of 푀 ⊕ 푁 preserving with higher cells matching relations on 퐾0. However, this the summands and trivial on 푁. Thus the group of auto- is difficult because of the nature of the relation on 퐾0: it morphisms we are concerned with is 퐺퐿(푅), the group of is a three-term relation. Adding higher cells between 0- infinite invertible matrices with entries in 푅 which are not cells can impose two-term relations, setting things equal equal to the identity only within a finite region. This also to one another, but it is much more difficult to impose partially captures the necessary stability. a three-term relation. One way to attempt this is to add Since composition of paths of isomorphisms matches points labelled by formal sums of projective 푅-modules. multiplication in 퐺퐿(푅), we have a homomorphism 푝 ∶ However, the more fruitful approach is to shift perspec- 퐺퐿(푅) 퐾1(푅). It turns out that stability also implies tive: a three-term relation can be represented by a triangle; that 푝 factors through 퐺퐿(푅)ab. Indeed, an element in thus constructing a space where arcs represent projective 푅- 퐺퐿(푅) can be represented by a finite matrix 퐴 by picking modules and triangles represent relations is a more natural a preimage under an inclusion 퐺퐿푛(푅) 퐺퐿(푅). Be- approach. As this produces a space whose 휋1 is 퐾0(푅), we cause of stability, for a given finite matrix 퐴, we must have end our construction by taking a loop space. One extra ad- vantage of this approach is that we no longer need to worry

4 In fact, historically speaking, we are doing this backwards. The definitions of 퐾0, 퐾1, and 퐾2 came first; they were then observed to fit into such an exact sequence, andtheideaof thinking of them as the homotopy groups of a space came later.

1038 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7 about constructing “negatives” of 푅-modules: since each We can add a 3-cell labelled by each triple of projective 푅- module is represented by a loop, its negative is simply the modules via a gluing map corresponding to this equation. same loop, traversed backwards. There are now relations that are imposed by four-term fil- We begin the construction with a single point. We then tered 푅-modules, and five-term filtered 푅-modules, and so add an arc for every finitely generated projective 푅-module. on; we thus produce an infinite-dimensional CW complex We then add a cell for each relation. It is important to note that is called |푠•퐏푅|; as we shall see in the next section, this that we need a relation for every way of assembling 퐴 ⊕ 퐵 is a special case of a construction called the Waldhausen 푠•- from 퐴 and 퐵; the best way to keep track of these is to label construction [20, Section 1.4].6 To get the 퐾-theory with the relations by exact sequences correct 휋0 we take its loop space, shifting the homotopy groups back down a dimension: 0 퐴 퐴 ⊕ 퐵 퐵 0.

def We now have the correct 휋1, but our higher homotopy 퐊(푅) = Ω|푠•퐏푅|. groups are far too large: many of these exact sequences have relations between them. We just need to figure out To get the higher 퐾-groups we define what such relations may look like. Already the case when 푅 is a field is highly nontrivial. def 퐾푛(푅) = 휋푛퐊(푅). For example, 퐾∗(퐂) is known completely only for finite coefficients [22, Section VI.1]. Suppose that we are givena 푉 푎+푏+푐 By construction this gives the correct 퐾0. Proving that this vector space of dimension . We wish to decom- ab pose it into spaces of dimensions 푎, 푏, and 푐. We can do gives GL(푅) on 휋1 is more difficult; for details on this this by selecting a subspace of dimension 푎, then quotient- see [22, Exact Categories IV.8.6, Corollary IV.7.2]. ing to get a space of dimension 푏 + 푐, then picking a sub- Algebraic 퐾-theory is very mysterious: it is famously dif- space of that of dimension 푏 and quotienting for the space ficult to compute and is related to many important invari- of dimension 푐. Alternately, we could pick a subspace of ants. For example, the Quillen–Lichtenbaum conjecture dimension 푎+푏, quotient to get a subspace of dimension (proved by Vladimir Voevodsky) states that there should 푐, and then decompose the subspace of dimension 푎 + 푏. be a spectral sequence beginning at ´etalecohomology of These ways of getting a subspace decomposition should, Spec 푅 and converging to the 퐾-theory of 푅. Assuming in some real sense, be equivalent. the Kummer–Vandiver conjecture (which is a statement There are several ways of correctly encoding such equiv- about the class numbers of cyclotomic fields) [21, Theo- alences. The most general is to shift our perspective on ex- rem 1] the 퐾-theory of the integers was computed to be act sequences: instead of thinking of an exact sequence as a 퐾0(퐙) ≅ 퐙, 퐾1(퐙) ≅ 퐙/2, and way of decomposing 퐴⊕퐵 into 퐴 and 퐵, we instead think 5 ⎪⎧0 if 푛 > 0 and 푛 ≡ 0 (mod 4) of it as a filtered 푅-module 퐴 ⊕ 퐵 with only two filtered ⎪ ⎪ layers. The three-term relation then says that the whole ⎪퐙 ⊕ 퐙/2 if 푛 ≡ 1 (mod 8) and 푛 > 1 ⎪ ⎪ module is equal to the first filtration plus the quotient. ⎪퐙/푐푘 ⊕ 퐙/2 if 푛 ≡ 2 (mod 8) This gives us a method for inserting new relations: extend- 퐾푛(퐙) ≅ 퐙/8푑푘 if 푛 ≡ 3 (mod 8) ⎪⎨ ing the length of the filtration. Given a filtered 푅-module ⎪ ⎪퐙 if 푛 ≡ 5 (mod 8) 퐴1 ⊆ 퐴2 ⊆ 퐴3, the three-term relation can break down ⎪ ⎪ 퐴 퐴 ⎪퐙/푐푘 if 푛 ≡ 6 (mod 8) 3 in two different ways. First, we could break down 3 ⎪ as 퐴2 and 퐴3/퐴2 and break down 퐴2 as 퐴2/퐴1 and 퐴1. ⎩퐙/4푑푘 if 푛 ≡ 7 (mod 8), Alternately, we could break down 퐴3 into 퐴3/퐴1 and 퐴1, and then further break 퐴3/퐴1 as (퐴3/퐴1)/(퐴2/퐴1) ≅ where 푐푘/푑푘 is the Bernoulli number 퐵2푘/푘 in lowest terms 퐴3/퐴2 and 퐴2/퐴1. These two ways of decomposing 퐴3 and 푛 = 4푘 − 1 or 4푘 − 2. The most famous computa- should be equivalent, as both decompose 퐴3 into 퐴1, tion of algebraic 퐾-theory is still Quillen’s original compu- 퐴2/퐴1, and 퐴3/퐴2. Such a filtered module should there- tation of the 퐾-theory of finite fields [14], which has since fore impose a four-term relation strongly resisted generalizations. More modern computa- tional techniques use close approximations to 퐾-theory, [퐴 ⊆ 퐴 ] + [퐴 ⊆ 퐴 ] 1 2 2 3 such as 푇퐻퐻 and 푇퐶; for more on this see for example = [퐴2/퐴1 ⊆ 퐴3/퐴1] + [퐴1 ⊆ 퐴3]. [9].

5We need a little bit more structure to make this work correctly; instead of taking just a fil- tration 푀 ⊆ ⋯ ⊆ 푀 we must take a sequence 푀 ⋯ 푀 of inclusions. In 0 푛 0 푛 6 ≅ This construction is one of two constructions in Waldhausen’s paper, called the 푠•- particular, isomorphisms 푀0 푀1 are allowed, and different choices of isomorphism construction and the 푆•-construction. The difference between them is that the 푆•- are assigned to different cells. construction can also incorporate a notion of “weak equivalence.”

AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1039 Intermezzo: Categories Representing Spaces 퐵 • The close connection between and cat- 푓 ℎ푔 egory theory is often presented historically: many of the 푔 fundamentals of category theory were developed by homo- topy theorists in connection with homology and cohomol- 퐴 • ℎ푔푓 • 퐷 ogy. Here we pursue a different approach illustrating the importance of categories in homotopy theory, with the fol- 푔푓 ℎ lowing slogan: • 퐶

Categories give algebraic models for homotopy Continue likewise for all 푛-tuples of composable types of spaces, analogously to the way that rings morphisms.7 In the case when the category is obtained of functions give algebraic models for geometric from a group 퐺 this construction produces the usual clas- objects. sifying space 퐵퐺 of the group. In the case when the cat- egory comes from a finite partial order (푃, ≤), this pro- This point of view clarifies why category theory is so inti- duces the abstract simplicial complex given by the set of mately intertwined with homotopy theory. To substanti- finite totally ordered subsets of 푃. Thus, for example, if ate the claims above, let us explore the basic constructions. 푃 = {푎1, 푎2, 푏1, 푏2} with the ordering given by 푎푖 ≤ 푏푗 This is a very informal introduction; for a more complete for 푖, 푗 = 1, 2, the category obtained from 푃 is drawn on treatment see, for example, [16]. the left, and its classifying space is the circle on the right: A category is a collection of objects, together with a col- 푏2 lection of morphisms between objects. A morphism has a 푏2 source and a target; a morphism with source 퐴 and target 푎1 푎2 퐵 is written 푓 ∶ 퐴 퐵. Morphisms have an operation 푎1 푎2 of composition: given a morphism 푓 ∶ 퐴 퐵 and a mor- 푔 ∶ 퐵 퐶 푔 ∘ 푓 ∶ 퐴 퐶 phism there is a morphism . 푏1 푏1 Some examples of categories: In the slightly more complicated case when 푃 = {푎1, 푎2, 푏1, 푏2, 푐1, 푐2} with the ordering given by 1. The category 퐒퐞퐭 has sets as objects and functions as 푎푖 ≤ 푏푗 ≤ 푐푘 for 푖, 푗, 푘 = 1, 2 the category obtained morphisms between sets (with the usual composi- from 푃 is drawn on the left and its classifying space is the tion). One can similarly define categories of groups sphere on the right: and group homomorphisms, topological spaces and 푏2 continuous maps, smooth manifolds and smooth 푏2 maps, etc. 2. Suppose that we are given a partial order (푃, ≤). We 푐1 can define a category with objects the elements of 푃, 푐1 푎1 푎2 푎1 푎2 and with exactly one morphism 푎 푏 if 푎 ≤ 푏. 푐2 Composition is well-defined by transitivity of ≤. 푐2 3. Let 퐺 be a group. We define a category with one object ∗ ∗ ∗ , and with the set of morphisms given by 푏1 푏1 the elements of 퐺. Composition is the multiplication |푠 퐏 | in 퐺. Comparing this to the construction of • 푅 in the pre- vious section, we see that it is almost directly analogous to the construction of the classifying space of a category. Given a category 풞 for which objects form a set, we can However, the 푠•-construction has 푛-simplices labelled by define the classifying space |풞|. Take a 0-cell for every ob- compositions with indexing 1, rather than 0; for example, ject in the category. For every morphism 퐴 퐵 attach a morphism gives a 1-simplex in the classifying space of a a 1-cell between the 0-cell for 퐴 and the 0-cell for 퐵. For category, but a 2-simplex in the 푠•-construction. We thus every pair of composable morphisms 푓 ∶ 퐴 퐵 and need an extra face to each simplex; this is given by taking 푔 ∶ 퐵 퐶 attach a 2-cell along 푓, then 푔, and then back- 퐴1 ⋯ 퐴푛 to 퐴2/퐴1 ⋯ 퐴푛/퐴1. wards along 푔 ∘ 푓. For every triple of composable mor- phisms 푓 ∶ 퐴 퐵, 푔 ∶ 퐵 퐶, ℎ ∶ 퐶 퐷 attach 7This is somewhat of an oversimplification, as we must deal with identity maps (and mor- 3 phisms that compose to identity maps) in a special way. This construction is the geometric a -cell by thinking of it as a tetrahedron and gluing it in realization of the nerve of the category; for a more detailed introduction see [16, Chapter along the triangles shown below: 6], as well as [10, Section 1.1].

1040 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7 Before we end this section, let us spend a little time con- by submodules (and quotients); thus we begin by consid- sidering how the categorical structure can help model the ering the disjoint union of finite sets. By analogous logic to homotopy class of a space. When working with categories, the section “Algebraic 퐾-theory,” this leads to 퐾0(퐅퐢퐧퐒퐞퐭) we do not work with categories up to isomorphism; in- ≅ 퐙, with the element 푛 representing a set of size 푛, and stead, we work up to equivalence. The idea here is that it 퐾1(퐅퐢퐧퐒퐞퐭) ≅ 퐙/2 representing the abelianization of should not matter how many different isomorphic mod- the group of permutations: the sign. Our hope is that the els of the same object exist in a category. For example, we 푠•-construction should give a construction of a 퐾-theory can consider the following three categories: space that agrees with these computations. 1. The category whose objects are finite subsets of 퐑, with We consider the simplices in the 푠•-construction. We functions as its morphisms. take an 푛-cell for each filtered finite set 퐴1 ⊆ 퐴2 ⊆ ⋯ ⊆ 2. The category whose objects are finite subsets of 퐙, with 퐴푛 and, thinking of it as an 푛-simplex, want to attach functions as its morphisms. it to the cells labeled 퐴1 ⊆ ⋯ ⊆ 퐴̂푖 ⊆ ⋯ ⊆ 퐴푛, as 3. The category whose objects are the sets {1, … , 푛} for well as (퐴2/퐴1) ⊆ ⋯ ⊆ (퐴푛/퐴1)—but what should 푛 ≥ 0, with functions as its morphisms. we take instead of /? The analog of taking quotients of modules is takings quotients of sets. In order for our 퐾0 All of these should model “the category of finite sets,” and = thus should all behave the same way, despite the fact that to work correctly, we need the 2-cell labeled by 퐴 ⊆ 퐴 each one restricts to a subset of the objects of the previous to be attached to 퐴, 퐴, and ∅, to represent the relation ones. This is the notion of equivalence of categories, and [퐴] = [퐴] + [∅]. However, if we take the quotient of it behaves analogously to the way that homotopy equiv- finite sets, we get [퐴] = [퐴]+[singleton], which would alences can “thicken” spaces by, for example, replacing a imply the relation |퐴| = |퐴|+1 inside 퐙; clearly not what circle with a solid torus. is intended. We can also use the algebraic structure of a category to To solve this problem we take a very simple approach: understand the homotopy type of its classifying space. For we simply take complements instead of quotients. We la- example, a category with an initial object (an object with bel the last face by a unique morphism to every other object in the category, (퐴 ∖ 퐴 ) ⊆ ⋯ ⊆ (퐴 ∖ 퐴 ), such as the empty set in 퐒퐞퐭) has a contractible classify- 2 1 푛 1 ing space: the initial object behaves like a “cone point,” which now gives the correct relations. We call this construc- connecting uniquely to everything else, and allows us to tion the 푠•̃ -construction. For a more detailed exploration retract every point in the classifying space to it. For those of this construction, see [7]. We can then define readers familiar with , we state the follow- def ing theorem, which is a generalization of this observation: 퐊(퐅퐢퐧퐒퐞퐭) = Ω|푠•̃ 퐅퐢퐧퐒퐞퐭|. By the Barratt–Priddy–Quillen Theorem, the 퐾-theory of 퐹 ∶ 풞 풟 ∶퐺 Theorem 3. Given a pair of adjoint functors finite sets is the space 푄푆0 (the 0-space of the sphere spec- the induced maps on the classifying spaces give mutually inverse 8 trum), whose homotopy groups are the stable homotopy homotopy equivalences. groups of spheres (the calculation of which is one of the In fact, it turns out that all homotopical behaviors of fundamental unsolved problems in algebraic topology to- topological spaces can be modeled by categories; this ob- day). servation is due to Thomason [19]. This sets up an interesting parallel. The integers are de- fined as a group completion of finite sets, with multipli- Combinatorial 퐾-Theory cation induced by Cartesian product of sets. The ring of When we constructed algebraic 퐾-theory we used almost integers is the fundamental object in number theory and none of the algebraic information about 푅; all we used is commutative algebra. Under 퐾-theory, the finite sets (with data about how to include and quotient 푅-modules. Thus a multiplicative structure induced by Cartesian product) this same kind of construction can be used with more geo- are related to the sphere , which is a fundamen- metric objects—such as varieties or polytopes—replacing tal object in homotopy theory and derived algebraic con- modules to construct a 퐾-theory of a more combinatorial structions. This illustrates that 퐾-theory is intrinsically in- nature. tertwined with the foundations of algebraic topology, pro- Let us consider the 푠•-construction again and attempt viding further evidence that our construction was well cho- to create a 퐾-theory for finite sets. Finite sets are made up sen. of their subsets, in the same way that modules are made up We can apply the 푠•̃ -construction more generally by looking at other types of objects that have complements. 8 This theorem can be proved as an application of [15, Theorem A]. It can also be proved 퐺 directly from the definition of the nerve; for a quick introduction to simplicial setsand For example, we can consider finite -sets (or almost-finite nerves, see for example [17]. 퐺-sets for a profinite group 퐺), polytopes in a uniform

AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1041 geometry, 푂-minimal structures, or varieties. In each of filtration on the space 퐊(퐕퐚퐫푘), we can extract informa- these the construction produces a 퐾-theory in which el- tion about the filtration on 퐾0(퐕퐚퐫푘). ements of 퐾0 classify (stable) scissors congruence invari- From our construction we see that the map ≤푛−1 ≤푛 ants. (For a more in-depth discussion of such examples 퐊(퐕퐚퐫푘 ) 퐊(퐕퐚퐫푘 ) induces an inclusion of and their 퐾-theory, see [7, 24].) spaces, which allows us to compute the cofiber of this map Let us investigate the case of varieties more closely. We as a spectrum.10 (For a much more detailed discussion of take the category whose objects are varieties over a field 푘, this, see [8, 23, 24].) In fact, we have the following: with closed immersions as morphisms. The 푠•̃ -construction gives a space 퐊(퐕퐚퐫푘) where 퐾0(퐕퐚퐫푘) Theorem 4 ([23, Theorem A]). The cofiber of the inclusion ≤푛 ≤푛−1 is the free abelian group generated by varieties, modulo 퐊(퐕퐚퐫푘 ) 퐊(퐕퐚퐫푘 ) is equivalent (as a spectrum) Σ∞(퐵BiratAut(푋) ) 퐵 the relation that to ⋁[푋]∈퐵푛 + . Here, 푛 is the set of bi- rational isomorphism classes of varieties of dimension 푛, [푋] = [푌] + [푋 ∖ 푌] BiratAut(푋) is the group of birational automorphisms of 푋, ∞ whenever 푌 is a closed subvariety of 푋. This is the Σ takes the suspension spectrum associated to a space, and ⋅+ adds a disjoint basepoint. Grothendieck ring of varieties. To construct 퐊(퐕퐚퐫푘) we can again use the Waldhausen 푠•̃ -construction. For the in- In particular, this means that, morally speaking, the clusions we use closed embeddings; for the complements 퐊(퐕퐚퐫 ) 9 spectrum 푘 is assembled out of classifying spaces we use the complement of the image of the embedding. of birational automorphism groups. As a consequence of Analogously to the case of finite sets defined above, this this we can conclude that all elements in the kernel of 휄푛 produces a space and we can define must be in the form [푋∖푈]−[푋∖푉] for some birational def automorphism 휑 ∶ 푋 푋 represented by an isomor- 퐊(퐕퐚퐫 ) = Ω|푠̃ 퐕퐚퐫 |. 푘 • 푘 phism 푈 푉. In this way the automorphism invariants The higher 퐾-groups encode invariants of living in 퐾1 control what can happen in 퐾0(퐕퐚퐫푘); the piecewise-automorphisms. higher structure of 퐊(퐕퐚퐫푘) is fundamentally intertwined In previous examples, the calculation of 퐾0 was sim- with its 0-level structure. ple, and it was the higher 퐾-groups that gave interesting invariants related to automorphisms. However, for vari- Coda: The Birds-Eye Perspective eties, even the calculation of 퐾0 is extremely nontrivial; it The focus of this article has been on problems arising in turns out the higher homotopy of 퐊(퐕퐚퐫푘) can create new algebraic or geometric contexts: vector bundles, projective tools for the analysis of 퐾0. modules, polytopes, etc. In each context, the moral of the Consider the filtration by dimension on the set of va- story was that for each such problem there is a 퐾-theory rieties. For any dimension 푛, we can construct a group space, whose homotopy groups classify higher invariants. ≤푛 In fact, there is an even more general approach to 퐾- 퐾0(퐕퐚퐫푘 ) generated by varieties of dimension at most 푛 (with the same relation as above). We can then filter theory. Any time that a problem can be modeled by a cat- 풞 11 퐾0(퐕퐚퐫푘) by setting the 푛-th filtration to be the image egory with an operation (such as finite sets with iso- ≤푛 morphisms as morphisms, and disjoint union as the op- of the natural map 휄푛 ∶ 퐾0(퐕퐚퐫푘 ) 퐾0(퐕퐚퐫푘). Un- fortunately, since we do not know that 휄푛 is injective, we eration) we can construct a topological monoid, since the ≤푛 classifying space |풞| of the category inherits an operation cannot use information about 퐾0(퐕퐚퐫푘 ) to learn about from 풞. The 퐾-theory is, morally speaking, the group com- 퐾0(퐕퐚퐫푘). Here is where having a space, rather than a set, is crucial. pletion of this monoid. There is an interesting theorem of A filtration on a space does not need to produce a filtration McDuff–Segal: on the connected components. Consider, for example, the Theorem 5 ([13]). For a topological monoid 푀, the 퐙 ⊆ 퐑 two-step filtration of the integers sitting inside the topologically correct group completion of 푀 is Ω퐵푀; here, 퐵푀 reals; the integers have infinitely many connected compo- is a generalization of the delooping of 푀. nents, while the reals only have one, but the filtration is topologically well-defined. In fact, the filtration quotient For a category 풞 with operation, it therefore makes is a bouquet of circles, which in its fundamental group en- sense to define def codes the data that the filtration joins infinitely many con- 퐊(풞) = Ω퐵|풞|. nected components into one. We can therefore hope that, if we can compute the associated grade of the dimension 10For those unfamiliar with spectra, it is possible to mentally replace “spectrum” with “space” in the following discussion. This will not produce correct mathematics, but will be 9As in the section “Algebraic 퐾-theory,” we need to remember the particular immersion close enough to explain the narrative. that was chosen for each cell in the construction. 11More formally, a symmetric monoidal structure.

1042 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7 This formula describes 퐊(풞) as a group completion, which [7] Campbell J. The 퐾-theory spectrum of varieties. is defined by its universal properties. Likewise, more gen- arXiv:1505.03136. eral 퐾-theory can be defined via a universal property (see [8] Campbell J, Zakharevich I. CGW-categories. arXiv: [4]). 1811.08014. [9] Dundas BI, Goodwillie TG, McCarthy R. The local struc- Unfortunately, this perspective comes with a severe ture of algebraic K-theory, Algebra and Applications, vol. 18, drawback: it is almost impossible to work with. Generally, Springer-Verlag London, Ltd., London, 2013. MR3013261 when constructing 퐾-theories, we wish to be able to prove [10] Goerss PG, Jardine JF. Simplicial homotopy theory, Progress things about them: to compute the higher 퐾-groups, to re- in Mathematics, vol. 174, Birkhäuser Verlag, Basel, 1999. late them to other forms of 퐾-theory, to compute the ho- MR1711612 motopy fibers or cofibers of maps between 퐾-theories, and [11] Hatcher A. Vector bundles and 퐾-theory. https://www. so on. The definition of 퐾-theory given by the McDuff– math.cornell.edu/~hatcher/VBKT/VBpage.html Segal theorem was not designed for such functionality, and [12] Hatcher A. Algebraic topology, Cambridge University thus (unsurprisingly) is not well-suited for such analysis. Press, Cambridge, 2002. MR1867354 [13] McDuff D, Segal G. Homology fibrations and the This is because the 퐵 functor in the definition is extremely “group-completion” theorem, Invent. Math., no. 3 difficult to work with, and is particularly unsuitable forcal- (31):279–284, 1975/76, DOI 10.1007/BF01403148. culations. MR0402733 The solution to this is often to find another construc- [14] Quillen D. On the cohomology and 퐾-theory of the tion—such as the 푠•-construction [20], Quillen’s 푄-con- general linear groups over a finite field, Ann. of Math. (2) struction [15], or the 퐵-construction for Γ-spaces [18]— (96):552–586, 1972, DOI 10.2307/1970825. MR0315016 for the 퐾-theory space. While the McDuff–Segal theorem [15] Quillen D. Higher algebraic 퐾-theory. I, Algebraic 퐾- gives a clean and simple definition, these constructions theory, I: Higher 퐾-theories (Proc. Conf., Battelle Memo- come with many tools that one can use to analyze their rial Inst., Seattle, Wash., 1972); 1973:85–147. Lecture Notes in Math., Vol. 341. MR0338129 results. Thus the study of 퐾-theory often sits awkwardly [16] Riehl E. Category theory in context, Aurora: Dover Mod- between the formal and the practical, trading off simplic- ern Math Originals, Dover Publications, 2017. https:// ity for usefulness and looking for beauty in between. books.google.com/books?id=6B9MDgAAQBAJ [17] Riehl E. A leisurely introduction to simplicial sets. ACKNOWLEDGMENT. The author would like to thank www.math.jhu.edu/~eriehl/ssets.pdf. Ilya Zakharevich for extensive edits and suggestions [18] Segal G. Categories and cohomology theories, Topology which massively improved the quality of exposition. (13):293–312, 1974, DOI 10.1016/0040-9383(74)90022- 6. MR0353298 The author would also like to thank the two anony- [19] Thomason R. Cat as a closed model category, Cahiers mous referees for their time and assistance in tightening Topologie G´eom. Diff´erentielle, no. 3 (21):305–324, 1980. up the exposition even further. MR591388 [20] Waldhausen F. 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AUGUST 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1043 Inna Zakharevich Credits Opening image is courtesy of the author. A celebration of Author photo is courtesy of Cornell University. AMS members! Join us on Wednesday, October 16, 2019 as we honor our AMS members via “AMS Day”, a day of specials on AMS publications, membership, and more! Stay tuned on Facebook, Twitter, and member emails for details about this exciting day. Spread the word about #AMSDay today!

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1044 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 7