Chromatic Stable Homotopy Theory and the AHSS

Now that we’ve all gotten over the excitement surrounding the new iPad, I wanted to talk about something else which I actually find very exciting (unlike the iPad). This semester Jacob Lurie is giving a course on Chromatic Homotopy Theory. This is a beautiful picture which relates algebraic topology and algebraic geometry. Hopefully with Jacob at the helm we’ll also see the derived/higher categorical perspective creeping in. This seems like a great opportunity the learn this material “in my heart”, as my old undergraduate advisor used to say.

And with most of our principal bloggers distracted by MathOverflow, it also seems like a good time to experiment with new media. So here’s the plan so far:

  • During lectures I’m going to be live-TeXing notes, which I’ll flush out and post to my website. (Jacob’s also posting his own notes!)
  • In addition, I’ll try to post blog articles (like this one) about related topics or topics I find interesting/confusing.
  • There might be a little MO action thrown in for fun.
  • The offshoot is that today I want to talk a little about chromatic homotopy and about the Atiyah-Hirzebruch Spectral Sequence.

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    Concrete Categories

    In many introductions to category theory, you first learn the notion of a concrete category: A concrete category is a collection of sets, called the objects of the category and, for each pair (X,Y) of objects, a subset of the maps X \to Y. (There are, of course, axioms that these things must obey.) In a concrete category, the objects are sets, and the morphisms are maps that obey certain conditions. So the category of groups is concrete: a map of groups is just a map of the underlying sets such that multiplication is preserved. So are the category of vector spaces, topologicial spaces, smooth manifolds and most of the other most intuitive examples of categories.

    Using terminology from a discussion at MO, I’ll call a category concretizable if it is isomorphic to a concrete category. For example, \mathrm{Set}^{op} can be concretized by the functor which sends a set X to the set 2^X of subsets of X, and sends a map of sets f:X \to Y to the preimage map f^{-1}: 2^Y \to 2^X.

    At one point, I learned of a result of Freyd: The category of topological spaces, with maps up to homotopy, is not concretizable. I thought this was an amazing reflection of how subtle homotopy is. But now I think this result is sort of a cheat. As I’ll explain in this post, if you are the sort of person who ignores details of set theory, then you might as well treat all categories as concrete. My view now is that specific concretizations are very interesting; but the question of whether a category has a concretization is not. I’ll also say a few words about small concretizations, and Freyd’s proof.

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    Three ways of looking at a local system: Introduction and connection to cohomology theories

    Suppose we have a space X. There are a lot of ways to describe the cohomology of X — algebraic geometers will know about étale cohomology, crystalline cohomology and (algebraic) deRham cohomology; topologists would add singular cohomology and simplicial cohomology to the list. Recently, thanks in large part to MIT’s K-theory seminar, I’ve come to understand how to tie the first three together. Each cohomology theory comes from a different way of looking at local systems.

    Roughly, a local system consists of a vector bundle V on X, together with some additional data which gives isomorphisms between different fibers of X. In algebraic geometry, there are many different ways to make this idea precise.

    In this series of posts, I want to present several of these ideas in the case of a smooth manifold X. In each case, if you pursued the idea far enough, you would get to a major tool of modern algebraic geometry.

    I said above that a local system includes the data of isomorphisms between different fibers V_x and V_y of X. This data can depend on the choice of a path \gamma between x and y. Our three theories will be distinguished by how short the path \gamma is required to be.

    We could work with an arbitrary \gamma. This would lead to étale sheaves

    We could work with an infinitesimal \gamma (also known as an \infty-jet). This would lead to crystals 

    We could work with a \gamma which is so small that it becomes a tangent vector. This would lead to \mathcal{D}-modules

    These give rise to étale cohomology, crystalline cohomology and deRham cohomology. I should point out that, for modern applications, one usually wants to work on stacks, with singularities and in arbitrary characteristic. I won’t be addressing any of those issues; I just want to give the intuition behind each theory.

    In this post, I will explain how to build a cohomology theory, given a notion of local system. I will then follow up with three more posts, one for each of the specific approaches above.

    As usual, this series comes with a disclaimer: These are tools that are relevant to my work, but their inner workings are not my expertise. If you want to see experts discussing this sort of thing, it looks like that conversation is going on at Urs’ journal club.

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    Lurie on TFTs

    This morning Jacob Lurie posted a draft of an expository paper on his work (with Mike Hopkins) classifying extended (infinity, n)-categorical topological field theories and their relation to the Baez-Dolan cobordism hypothesis.

    Should make for some intersting bedtime reading…

    Live Blogging: AJ on Gromov-Witten theory of stacks

    Today AJ’s talking in the Grand Unified Seminar (representation theory, geometry, and combinatorics) on his this (joint work with his collaborator C. Teleman and his advisor E. Frenkel).  The title of the talk is “Gromov-Witten Theory for a point/C*.”  As AJ points out with delight (after the title was misintroduced without the mod C*, “It’s negative 2 dimensional!”

    The outline of the talk is:

    1. Gromov-Witten Invariants
    2. The stack pt./C*
    3. Integration on quotient stacks
    4. (the unfortunately named) admissible classes
    5. Bundles on nodal curves
    6. Invariants are well-defined

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    Generalized Homology Theories

    Recently there have been some comments on our requests page:

    Sander Kupers:

    Maybe you could explain a bit about elliptic cohomology and topological modular forms…

    and Thomas Riepe:

    I would be curious about learning more on:
    “… many constructions of classical algebra (eg, the theory of modular forms) are beginning to be seen to have deep homotopy-theoretic foundations.”…

    Since this is somewhat related to some of my research, I have been recruited volunteered to talk about these things. The problem is that this is a Huge subject and a difficult subject and there is no way to adequately represent it in blog form. On the other hand it is a beautiful subject, filled with lots of exciting tidbits. Why not give it a go anyway, right?

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    Reference Hunt I

    Does anyone know where the following useful facts were first proved? A lot of papers just say, "It is known that…" and I’d like to give proper attribution in some future work.

    Let A be an abelian group, and let Vect^A denote the monoidal category of A-graded complex vector spaces. Then:

    1. Equivalence classes of braided structures on Vect^A are classified by elements of H^4(K(A,2),\mathbb{C}^\times).
    2. H^4(K(A,2),\mathbb{C}^\times) also classifies \mathbb{C}^\times-valued quadratic forms on A.
    3. H^4(K(A,2),\mathbb{C}^\times) = H^3_{ab}(A, \mathbb{C}^\times), where the right side is "Eilenberg-MacLane abelian group cohomology" (defined in MacLane’s 1950 ICM address).

    There is an additional neat interpretation involving double loop maps and multiplicative torsors on A, but I don’t need that level of sophistication for the near future.

    Embedded TQFT?

    So, a subject rather near and dear to the hearts of many of my fellow co-bloggers is that of 1+1-dimensional TQFT: that is, of monoidal functors from the category of 1-manifolds with morphisms given by smooth cobordisms to the category of vector spaces over your favorite field k.

    There’s a rather remarkable theorem about such functors, which really deserves a post of its own for proper explanation, but I’ll spoil the surprise here.

    Any such functor associates a vector space A to a single circle, and to the “pair of pants” cobordism, it assigns a map m:A\otimes A\to A, which one can check is a commutative multiplication.

    Furthermore, the cap, thought of as a cobordism from the empty set to a circle gives a map i:k\to A, which gives a unit of this algebra. Thought of as a cobordism from the circle to the empty set, it gives us a map \mathrm{tr}:A\to k which we call the counit or Frobenius trace.

    Theorem. A commutative algebra with counit (A,\mathrm{tr}) arises from a TQFT if and only if \mathrm{tr} kills no left ideal of A.

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