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Attitudes of -Theory 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 mathematics 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 vector bundle 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 homotopy 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 suspension 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 quaternions 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.
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