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This page was the home of the **X**traordinary cohomology theory & **K**-theory **C**ollective **D**iscussion group^{*} for the fall 2011 semester.

All talks at Berkeley were on Berkeley time (i.e. they started 10 minutes late).

** 1:00-2:00 -- An introduction to string topology**

In 1999 Chas and Sullivan defined a product on the homology of *LM*, the space of unbased loops in a compact manifold *M*. This product turns *H _{*}(LM)* into a BV-algebra. We will start by describing this product. We will construct a "cacti operad" that acts on

references: R.L. Cohen & A. Voronov,

**2:30-3:30 -- What is elliptic cohomology?**

Elliptic cohomology is a relatively new field in algebraic topology that has sparked a great deal of interest and excitement (as well as deep conjectural relationships with quantum field theories). In this talk, I'll describe what an elliptic cohomology theory is, where it comes from, and why you should care.

*Aaron Mazel-Gee*

references: Lurie, *A survey of elliptic cohomology*.

** 1:00-2:00 -- Co/simplicial spectral sequences**

Simplicial complexes are much more useful and versatile than mere combinatorial models for topological spaces, the first context they're introduced in. The goal of this talk is to show how co/simplicial objects and their associated spectral sequences can be used to help organize your topological life generally. We will instantiate these objects in a variety of settings and compute some explicit examples to see what's what.

*Eric Peterson*

references: Lecture notes.

** 2:30-3:30 -- Stable homotopy theory**

We'll discuss what stable homotopy theory is, why we care about it, and to outline a few different (ultimately equivalent) constructions of the "stable homotopy category." Objects of this category are usually called "spectra." Some of them come from spaces, and all of them correspond to (extraordinary) cohomology theories. We'll discuss some abstract properties of spectra, and relate their category to categories we already care about (topological spaces and groups). For example, we can show that multiplication in cohomology and (when it exists) in homology comes from multiplication in the stable homotopy category. Poincare duality also has an interesting interpretation here before we pass to homology groups.

*Cary Malkiewich*

references (roughly in order of readability): Lecture notes. Schwede, *An Untitled Book Project on Symmetric Spectra*. Whitehead, *Generalized Homology Theories*. Adams, *Stable Homotopy and Generalised Homology*. Mandell, May, Schwede, & Shipley, *Model Categories of Diagram Spectra*. May & Sigurdsson, *Parametrized Stable Homotopy Theory*.

** 1:00-2:00 -- The homology of the connective covers of BU, part 5**

Last spring, we began a series of talks exploring the formal geometry that arises in the study of characteristic classes of complex vector bundles with certain extra structure. This is a rather winding story that we've gotten only halfway through, and so the purpose of this talk is to recollect what we've accomplished thus far. The ultimate goal at the end of this sequence is the following theorem of Ando, Hopkins, and Strickland: there exists a multiplicative map of spectra *MU<6>* → `tmf` called the "sigma-orientation", which on homotopy gives (the complex version of) the "Witten genus".

*Eric Peterson*

references: Jump to the first lecture in this series.

** 2:30-3:30 -- Atiyah duality of manifolds**

Atiyah duality is a statement about all manifolds *M* (not just oriented ones) that shares the same bed as Poincaré duality, Spanier-Whitehead duality, Alexander duality, and the Thom isomorphism. We'll disentangle these things and see what's always true for non-oriented manifolds. Then we'll spectrify and talk about what Atiyah duality says in the stable homotopy category, the natural setting for stable statements like this. We'll make a relative statement for manifolds with boundary (the analogue of Lefschetz duality).

If there's time, we'll state "parametrized Atiyah dualty". This lives on top of the mountain of parametrized stable homotopy theory, but when we walk back down, we get Poincaré duality with local coefficients (for non-oriented manifolds). We could also walk down a different side of the mountain and end up with a statement about string topology.

*Cary Malkiewich*

references: Cohen, *Multiplicative Atiyah Duality*. Cohen & Jones, *Umkehr Maps*. May & Sigurdsson, *Parametrized Stable Homotopy Theory*.

** 1:30-2:30 -- The Dold-Thom theorem**

The Dold-Thom theorem is a magic recipe for converting homology into homotopy. I will try to convince you of its usefulness by giving an application. I will also talk about ways to prove it, although I might not give a precise proof.

*Ilya Grigoriev*

** 3:00-4:00 -- From Morse theory to Bott periodicity, part 3**

Last semester, we laid the groundwork for Bott's original proof of his celebrated periodicity theorem. We now return to finish off the story. I'll start by reviewing the Morse theory of path spaces of manifolds; this can be made quite explicit for symmetric spaces, which include compact Lie groups as special cases. Then, we'll use what we've learned to run through Bott's incredibly satisfying proof that *Ω ^{2}U ~ U* (where

references: Jump to the first lecture in this series.

** 2:00-3:00 -- Twisted Poincaré duality**

I'll give a useful and powerful theorem that specializes to all of the variants of Poincaré duality (or at least the ones that still involve a manifold somehow). In particular, the manifold could be nonoriented and noncompact, and the cohomology theory could be extraordinary and twisted over the manifold. We'll introduce some machinery, give the statement, show how it implies lots of more classical theorems, and give a rough sketch of the proof. It indicates that the world of parametrized stable homotopy theory is a very powerful framework for making statements about manifolds and extraordinary cohomology theories.

*Cary Malkiewich*

** 3:30-4:30 -- The homology of the connective covers of BU, part 6**

Last time, we reviewed our progress thus far on describing Spec *H _{*}BU<2k>*, and we completed the last input calculation we'll need to finish the story. In this installment, we'll see that we've been measuring objects already known to be interesting to arithmetic geometers, and we'll borrow some constructions from their field to complete our obstruction framework.

references: Jump to the first lecture in this series.

** 1:30-2:30 -- Rational homotopy theory**

Rational homotopy theory is an example of algebraic topology doing what it does best: reducing geometric information about topological spaces to relatively easily computable algebraic structures. Indeed, the seminal work of Sullivan shows that the homotopy category of (simply-connected) topological spaces with torsion-free homology is equivalent to the category of (reduced) cocommutative differential graded coalgebras, which are very friendly objects! However, in this talk, I will take the `Koszul-dual' approach and instead describe the even more seminal work of Quillen in terms of the equivalence with (reduced) differential graded Lie algebras. Since I will surely not have time to go into much detail in any of the technical arguments, I will instead attempt to give a high-level overview of the conceptual ideas that go into each step of Quillen's argument and in particular why one should expect such a nice result over **Q** but not over other fields.

*Arnav Tripathy*

references: Quillen's paper, this mathoverflow question, and of course Lurie's DAG X and XIII (the last few suggested references are mostly in jest).

** 3:00-4:00 -- Nilpotence in stable homotopy theory**

*Aaron Mazel-Gee*

references: Devinatz, Hopkins, & Smith, *Nilpotence and stable homotopy theory I*. Hopkins & Smith, *Nilpotence and stable homotopy theory II*. Peterson, *Devinatz-Hopkins-Smith I, II, & III*.

** 2:30-3:30 -- Extraordinary homotopy groups**

In the Berkeley installments of the xkcd seminar, we've had several lectures focusing on the role of formal geometry in algebraic topology, and now it's finally time for some of this to spill over into Stanford. In this talk, we'll introduce the field of chromatic homotopy theory, which is where all the major advancements on the *π _{*}^{S}* problem have come from in the past 30+ years. Our express goal will be to study the Picard groups of the

references: Lecture notes.

At 4:00 we'll attend the Stanford topology seminar, where Constantin Teleman (of UC Berkeley) will talk about 2D gauge field theory via name-dropping: Gromov, Kapustin, Langlands, Rozansky, Witten.

** 11:00-12:00 -- The homology of the connective covers of BU, part 7**

*Eric Peterson*

references: Jump to the first lecture in this series.

** 1:00-2:00 -- The homology of the connective covers of BU, part 8**

*Eric Peterson*

references: Jump to the first lecture in this series.

** 11:00-12:00 -- 200-032 Lane History Corner -- Whitehead torsion and the s-cobordism theorem**

I will show how an algebraic obstruction, called Whitehead torsion, arises naturally when trying to carry out the proof of the h-cobordism theorem in the non-simply connected case. This leads to the statement, and half of the proof, of the s-cobordism theorem. I will then show that Whitehead torsion has a very nice interpretation in terms of the combinatorics of cell complexes, which will lead to the other half of the proof. (No prior knowledge of the h-cobordism theorem assumed.)

*Sam Nolen*

** 2:30-3:30 -- 383N Sloan Hall -- Dieudonné modules and the classification of formal groups**

Just as Lie groups can be understood through their Lie algebras, formal groups can be understood through their associated "Dieudonné modules". We'd like to mimic Lie theory, but over an arbitrary ring we don't have anything like the Baker-Campbell-Hausdorff formula at our disposal. Instead, we study the full group of formal curves through the origin of our formal group, along with the actions of some geometrically-flavored endomorphisms as well as some purely algebraic Frobenius endomorphisms. This turns out to be enough structure to give us an equivalence of categories between formal groups and Dieudonné modules.

In this talk, I'll remind you why topologists care about formal groups, introduce the various algebraic objects at play, and illustrate some rather striking classification results.

*Aaron Mazel-Gee*

references: Lecture notes. Ando, *Dieudonné crystals associated to Lubin-Tate formal groups*.

^{*} the xkcd group -- making xkcd not stand for nothing since 2010!