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POWER OPERATIONS 1. 2/13: Talk by Jacob Lurie This Is Motivation For

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POWER OPERATIONS 1. 2/13: Talk by Jacob Lurie This Is Motivation For POWER OPERATIONS Abstract. Notes from the spring 2014 \Thursday seminar" at Harvard. TEXed by Akhil Mathew, who takes full responsibility for all errors. 1. 2/13: Talk by Jacob Lurie This is motivation for what we're talking about in the Thursday seminar this semester. I'm going to start by talking about Dijkgraaf-Witten theory. Let G be a finite group, and fix η 2 Hn(BG; C×). If M n is an n-dimensional manifold, you can associate to it a number Z(M n). Namely, R ∗ X n P η Z(M n) = M jAut(P )j iso classes of G bundles P The numerator is as follows. G-bundles on M are given by maps M ! BG. Given a G-bundle P , identified with a map P : M ! BG, you get a class by pulling back a class in Hn(M; C×). You can then integrate this class and then come up with a class in C×. Example 1. If η = 1, then this is counting the number of G-bundles on M with multiplicity. In general, it is some twisted version of that count. This gives an invariant of n-manifolds, and this gives the top-level of a topological quantum field theory. If M is a manifold of dimension n − 1, then there is an analogous procedure that you can perform that you can use to perform a complex vector space. Again, you're going to think about all G-bundles on M, but instead of thinking of them one at a time, consider the classifying space of them. That is, consider Map(M n−1; BG). If I take this, there is a map : Map(M; BG) × M ! BG; given by evaluation. Moreover, we have a projection map p : Map(M; BG) × M ! Map(M; BG): If I start with η 2 Hn(BG; C×), pull back along the evaluation map to give a class ∗η in Hn(Map(M; BG)× × R ∗ 1 × M; C ). Then, consider the pushforward along p η 2 H (Map(M; BG); C ). Now H1 with coefficients in C× parametrizes local systems of 1-dimensional complex vector spaces. R ∗ So this class η classifies a local system LM on Map(M; BG). Earlier, we got a locally constant function | we got something in H0. We were able to sum it over the various connected components. So, we define n−1 Z(M ) = Γ(Map(M; BG); LM ): This is an invariant of any n − 1-dimensional manifold; it's a finite-dimensional complex vector space. The mapping space Map(M; BG) has finitely many connected components and each connected com- ponent is the classifying space of a finite group (the automorphism group of a finite G-bundle). So giving a one-dimensional local system means giving a bunch of characters of a bunch of finite groups and taking global sections means taking invariants and adding them up. How are these (n − 1)-dimensional invariants related to these n-dimensional invariants? To think about this, consider an (n − 1)-manifold M n−1, another N n−1, and an n-manifold B giving a bordism between the two. In this case, you can look at Map(B; BG), and that projects onto Map(M; BG) × Map(N; BG). So we get maps f : Map(B; BG) ! Map(M; BG); g : Map(N; BG): Date: May 11, 2014. 1 2 POWER OPERATIONS ∗ ∗ We get local systems LM ; LN on the two targets and we can pull back to get f LM ; g LN on Map(B; BG). These are related. These local systems came from pulling back certain cohomology classes on BG and integrating them. Inside this manifold B, the fundamental cycles of M and N are homologous { B itself is a boundary between them. So in B, integrating along M and integrating along N are ∗ ∗ identified. So we should get f LM ' g LN . If you do this process at the level of cycles, you get a ∗ ∗ preferred choice of isomorphism f LM ' g LN . So, given a section of LM , I can pull it back and push it forward to get a section of LN . So, I get a map Z(B) Z(M n−1) ! Z(N n−1); and when M; N are empty, then the two vector spaces are canonically C, so the map is multiplication by the number we encountered earlier. More precisely: n−1 0 Definition 1. Z(M ) = H (Map(M; BG; LM ). (There is no higher cohomology since we are over C.) Now there is a good way of pulling back cohomology classes, but not pushing forward. So, crucial to this discussion was the fact that homology and cohomology were the same | so I could push forward either. At the level of vector spaces, this comes from the fact that if V is a vector space over C with G an action of a finite group G, then V ' VG canonically. Two years ago in this seminar, we proved that there is a similar phenomenon of homology and cohomology giving you the same answer in a different context. Let X be a space which has finitely many finite homotopy groups. A space like BG is such a space. Moreover, Map(M; BG) is such a space. Let L be a local system on X of K(n)-local spectra. Then we can talk about C∗(X; L) and C∗(X; L) (\chains" and \cochains" of L on X). Definition 2. C∗(X; L) is the homotopy limit over X of L.(L is a diagram of spectra parametrized by X.) Similarly, C∗(X; L) is the homotopy colimit of this diagram. The homotopy colimit is taken in the world of K(n)-local spectra. Two years ago, we discussed the fact that if X is as above, there is a canonical equivalence ∗ C∗(X; L) ' C (X; L): The slogan is that such a space, from the point of view of K(n)-local homotopy theory, behaves like a zero-dimensional space: homology and cohomology are canonically the same. A consequence is that you can run the same construction for Dijkgraaf-Witten theory. Let's restart the discussion. Let E be a Morava E-theory of height n. I'll review what these are a little later. This is some kind of cohomology theory, and a theorem of Goerss-Hopkins-Miller implies that it is a well-structured cohomology theory. E has an essentially unique structure of an E1-ring. There is a multiplication in E-cohomology, but somehow it satisfies all the commutativity that you would want up to coherent homotopy. 1 The upshot is that it makes sense to consider GL1(E) ⊂ Ω E, and this is the subspace of those 1 connected components in Ω E which are units at π0. That's a space that you can assign to any kind of multiplication on a spectrum. You just need a way of picking out the units in π0. But if you've got one of these fancy multiplications, then this GL1(E) is actually an infinite loop space. Any operad that acts on E as a spectrum will act on its units as a space. So GL1(E) gets an action of an E1-operad, 1 so you can deloop it as many times as you want. Therefore, you can write GL1(E) = Ω gl1(E) where gl1(E) is a spectrum. n Now it makes sense to talk about cohomology. Start with a cohomology class ηn 2 H (BG; gl1(E)). By definition, this is n −n H (BG; gl1(E)) = [BG; Ω GL1(E)]: Starting with η, we can mimic all the previous constructions. At various points, we need to invoke the ambidexterity theorem to avoid distinguishing between homology and cohomology. This means that we can do the following: POWER OPERATIONS 3 (1) To every closed n-manifold M, assign Z(M). Before it was a complex number; now it's a number in π0E. (2) Given an n − 1-manifold, assign to it Z(M n−1), which is going to be an E-module spectrum. These are related to each other in the previous way { functoriality for bordisms. All of this is motivation; it's not going to be what the seminar is going to be about. But I'm telling you that once you choose such an element η, you can construct a bunch of manifold invariants. If you thought that this was interesting and wanted to get manifold invariants, you might ask what types of η to choose. Now we're getting to the subject matter of this seminar. ∗ Question: What can we say about H (BG; gl1(E))? That's sort of like asking what this spectrum gl1(E) looks like. Maybe I don't care about everything here. I just want to map reasonably simple things into it, like the classifying spaces of finite groups. Let's start with this question of what gl1(E) looks like. How might you answer this question? It's a spectrum, so it's got homotopy groups. What are its homotopy groups? That's pretty easy. Note that gl1(E) is the connective delooping of GL1(E), so no homotopy groups in negative degrees, by definition. The zeroth space determines the other homotopy groups, so we get 8 0 if ∗ < 0 <> × π∗gl1(E) = (π0E) if ∗ = 0 : > :π∗E if∗ > 0 Just knowing the homotopy groups of a spectrum is not necessarily so useful if we want to know, for instance, how to map into it.
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