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Harvard-MIT Homotopy Theory Seminar Summer 2022

Introduction to Stable and Chromatic Homotopy Theory

Overview:

Chroma is a joint Harvard-MIT summer seminar primarily focused at undergraduates interested in algebraic topology and homotopy theory. The seminar is now over, but you can see the talks page for abstracts and slides! We may be coming back in 2023, so stay tuned!

This summer, we will try to define the stable homotopy category, understand its desirable properties for computational and abstract homotopy theory, and present some results from chromatic homotopy theory. Roughly, we will follow Ravenal’s Nilpotence and Periodicity in Stable Homotopy Theory, but will spend some time reviewing preliminaries to this book. The goal is to develop an intuituion and appreciation for the machinery of stable homotopy theory without getting bogged down by technicalities and computation.

What is “Stable Homotopy Theory”?

The main objects of study in classical algebraic topology are (sufficiently nice, e.g. CW) spaces. The general perspective is that while understanding a space (say, up to homeomorphism) is too hard in general, we can attempt to look at spaces up to homotopy equivalence, which really means going from trying to understand the space \(\text{Maps}(X,Y)\) to considering homotopy classes of maps, denoted by \([X,Y]\) (these are equivalent by the Yoneda lemma).

But even here we are stuck, because in general \([X,Y]\) might just be a set. However, if \(X\simeq \Sigma X'\) (assume everything is based), then we can identify \([X,Y]\) with \(\pi_1\text{Maps}(X',Y)\), which is a group. If \(X\) is a double suspension, then this becomes an abelian group. Thus, we might want to look at the tower

\[[X,Y] \xrightarrow{\Sigma} [\Sigma X, \Sigma Y] \xrightarrow{\Sigma} [\Sigma^2 X, \Sigma^2 Y] \xrightarrow{\Sigma} \cdots\]

as a kind of approximation of \([X,Y]\), which is a priori just a set, by things which do have tangible algebraic structures to latch onto. From this point of view, what we really should be studying, as in the thing which we have the best chance of getting a handle on, is the direct limit of this tower, which we call the stable homotopy classes of maps \([X,Y]_s\), because these are precisely the maps \(f\colon X\to Y\) which are invariant under suspension (this is what it means to be “stable”). Our takeaway should be this ethos to what we call “stable homotopy theory”:

The simplest things are stable things

Let’s take this analogy one step further: why is topology hard and algebra easy? (If this wasn’t at least somewhat, then we’d hardly want to study topological spaces via algebraic invariants!) In some sense, it’s because \(\mathbf{Ab}\), the category of abelian groups, is really nice (e.g. it’s abelian, closed symmetric monoidal, etc.), while \(\mathbf{Spaces}\) (or even \(\text{ho}\mathbf{Spaces}\)) aren’t, well, anything much in general. By our stated ethos, one way to think about the transition from classical homotopy theory to stable homotopy theory is to think of \(\mathbf{Spaces}\) not being “nice” as a bug instead of a feature.

To fix this, what we want is a new category, \(\mathbf{SHC}\), called the “stable homotopy category,” which is nicer in the sense of giving us more algebraic structure to work with. In the stable category, the morphisms between objects given precisely by the stable classes of maps. The objects are called spectra. What are the spectra? We want these to be “spaces which are suspension-invariant,” and while there are many ways of actually constructing the category of spectra (the homotopy category of which is precisely \(\mathbf{SHC}\)), a straightforward one is that a spectrum \(E\) is a sequence of spaces \(E_0, E_1, E_2, \ldots\) and structure maps \(\Sigma E_n \to E_{n+1}\).

A good source of examples of spectra is suspension spectra, where we start with a space \(X\), set \(E_n=\Sigma^n X\), and make the structure maps the identity. Applying this to the 0-sphere, we get the sphere spectrum \(\mathbb{S}\), the homotopy groups of which (in the stable category) are the stable homotopy groups of the spheres. Another good source of examples are generalized cohomology theories — a result called Brown Representability says that given any spectra \(E\), the axioms of a spectra are exactly what we need for the functor \(E^n(X) = [Y,E_n]\) to be a generalized cohomology theory; i.e. a functor from \(\text{ho}\mathbf{Spaces}\) to \(\textbf{Ab}\) satisfying the Eilenberg-Steenrod axioms (if this reminds you of Eilenberg-Mac Lane spaces, you’ll be delighted to know that the sequence \(K(A,0),K(A,1),K(A,2),\ldots\) forms a spectrum called \(HA\), which represents ordinary cohomology with \(A\) coefficients).

Historically, much of the formalism of stable homotopy and spectra arose when Frank Adams was working on the Hopf invariant one problem, but there are echos of this idea throughout classical homotopy theory, such as via the Freudenthal suspension theorem or the aforementioned connection between spectra and generalized cohomology theories. On the other hand, modern perspectives on spectra — which are often phrased in the language of \(\infty\)-categories — treat spectra as the final step in the effort to make working in \(\text{ho}\mathbf{Spaces}\) more like doing algebra, and approach stable homotopy theory from the lens of “homotopy-theoretic algebra.” Our goal for Chroma is to take a middle ground — we’ll start by defining the stable category and understanding just what makes it such a nice place to do homotopy theory, and then apply this background to understanding some classical results, such as the nilpotence theorem.

Schedule:

Date	Speaker	Topic	References
May 27th		Organizational Meeting
June 2nd	Dora Woodruff	Outline & Main Results Intro to Spectra	Orange Book Ch. 1 Barnes-Roitzheim Ch. 1
June 9th	Keita Allen	Categories of Spectra Monoidal Structures	Barnes-Roitzheim Ch. 2,5,6
June 16th	Jonathan Buchanan	Bundles, Thom spaces Thom Isomorphism Theorem	Miller Ch. 6,8
June 23rd	Merrick Cai	Bousfield Localization	Barnes-Roitzheim Ch. 7 Ravenel Ch. 7
Week 5	Kenta Suzuki	Steenrod Algebra, Adams Spectral Sequence	Barnes-Roitzheim Ch. 2.5-2.6
Week 6	Charley Hutchinson	Formal Group Laws, Complex Orientations	Ravenel Ch. 3,4
Week 7	Rushil Mallarapu	The Ravenel Conjectures, Chromatic Filtration	Ravenel ‘84
Week 8	Rushil Mallarapu	Proof of the Nilpotence Theorem	Devinatz-Hopkins-Smith ‘88’