Chromotopy

Higher order Weil pairings

August 24th, 2011

The summer is coming to a close, and so I’ve been trying to summarize some of the things that have happened. For one, I did manage to successfully perform the computation of $(E_n)^\vee_* K(\mathbb{Z}/p^j, q)$ for all conceivable values of n, p, j, and q, and the whole thing did culminate, as expected, in an isomorphism of graded ring schemes

$\mathbb{C}\mathrm{P}^\infty_E[p^\infty]^{\wedge *} \cong K(\mathbb{Z}/p^\infty, *)_E.$

So, that’s pretty exciting. But some other things happened too, and here’s one I’ve been meaning to complain about for a while. For the most part, this idea of associating to a compactly generated space X the formal scheme $X_E := \mathrm{colim}_\alpha E^* X_\alpha$ , where $X_\alpha$ ranges over the compact subspaces of X, is a mechanism for introducing the language and ideas of algebraic geometry into algebraic topology. The first nontrivial example that I learned in this framework is the Ando-Hopkins-Strickland result (or, rather, the Cartier dual of what I’m about to say) that the formal scheme $BU\langle 2k \rangle_E$ corresponds to the kth symmetric power of the augmentation ideal of a “group ring” type object $C_0 \mathbb{C}\mathrm{P}^\infty_E$ , for all kinds of theories E and at least in the range $k \le 3$ . A big part of their proof was an explicit analysis of the schemes $C_k$ , especially in the case of k = 3, where chasing out the definition of $C^3 := \mathrm{Hom}(C_3 \mathbb{C}\mathrm{P}^\infty_E, \mathbb{G}_m)$ shows that it is the scheme of “cubical structures”.

In topology, there is a fiber sequence $K(\mathbb{Z}, 3) \to BU \langle 6 \rangle \to BSU$ , which for a p-local cohomology theory E gives at the very least a map

$\mathrm{Hom}(K(\mathbb{Z}/p^\infty, 2)_E, \mathbb{G}_m) = \mathrm{Hom}(K(\mathbb{Z}, 3)_E, \mathbb{G}_m) \leftarrow \mathrm{Hom}(BU\langle 6 \rangle_E, \mathbb{G}_m).$

But, at least for specific choices of E like the one in my opening unrelated paragraph, we know that $K(\mathbb{Z}/p^\infty, 2)_E \cong (B\mathbb{Z}/p^\infty_E)^{\wedge 2}$ , and so we expect the map above to mimic a map in algebraic geometry of the form

$\mathrm{Hom}((B\mathbb{Z}/p^\infty_E)^{\wedge 2}, \mathbb{G}_m) \leftarrow C^3(\mathbb{C}\mathrm{P}^\infty_E, \mathbb{G}_m),$

i.e., something that swallows cubical structures and outputs alternating biexponential maps. Such an object is known to the enterprising algebraic geometer: the Weil pairing. This object was crucially used by Mumford in his study of biextensions, and has received all kinds of press before and since.

To give you the dimmest sense of what this is, if you’re unfamiliar with it already, I’ll outline its construction. A biextension over a pair of group schemes G and H is a $\mathbb{G}_m$ -torsor over $G \times H$ , such that if you pick any point $g \in G$ or $h \in H$ , the pullback of the torsor along the inclusions $\{g\} \times H \hookrightarrow G \times H$ and $G \times \{h\} \hookrightarrow G \times H$ respectively is an honest split central extension of the base by $\mathbb{G}_m$ . Each of these central extensions has a controlling 2-cocycle determining the isomorphism type of the extension, and a “cubical structure” is a biextension for which $G = H$ , all these 2-cocycles are equal, and the 2-cocycle is $\Sigma_3$ -invariant. Beginning with a cubical structure $\mathcal{L}$ , we can consider the multiplication-by-p maps $p_L$ and $p_R$ in the left- and right-hand factors of the base respectively. These maps factor as

$\mathcal{L} \xrightarrow{i} \mathcal{L}^{\otimes p} \xrightarrow{\mu_L} \mathcal{L},$

and similarly for $p_R$ . These two maps $i$ and $\mu_L$ *themselves* factor as $\mathcal{L} \to p_* \mathcal{L} \to \mathcal{L}^{\otimes p}$ and $\mathcal{L}^{\otimes p} \to (p \times 1)^* \mathcal{L} \to \mathcal{L}$ . The two factoring maps $p_* \mathcal{L} \to \mathcal{L}^{\otimes p}$ and $\mathcal{L}^{\otimes p} \to (p \times 1)^* \mathcal{L}$ can be checked to be isomorphisms, and finally the composite

$(p \times 1)^* \mathcal{L} \to p_* \mathcal{L} \to (1 \times p)^* \mathcal{L}$

is the thing called the “Weil pairing”. Over the p-torsion in the base, at least, the two pullback torsors trivialize canonically, and so we can write down a formula for the action of the composite map in the $\mathbb{G}_m$ factor:

$e_n = \prod_{i=1}^{p-1} \frac{u(x_1, ix_1, x_2)}{u(x_1, ix_2, x_2)},$

where u is the controlling 2-cocycle. And, finally, the real punchline: this is exactly the map induced by the topological fiber $K(\mathbb{Z}, 3) \to BU\langle 6 \rangle$ .

Of course, the topological story doesn’t stop there. At the next stage in the Whitehead-Postnikov decomposition there’s another fiber sequence $K(\mathbb{Z}, 5) \to BU\langle 8 \rangle \to BU\langle 6 \rangle$ , and we can attempt to play a similar game by pplying the functor $\mathrm{Hom}(-_E, \mathbb{G}_m)$ to the fiber map. Again, assuming that E is a nice cohomology theory, the object $\mathrm{Hom}(K(\mathbb{Z}, 5), \mathbb{G}_m)$ is known: it’s the scheme of alternating, multiexponential functions on the p-divisible subgroup $\mathbb{C}\mathrm{P}^\infty_E[p^\infty]$ of the formal group associated to the complex orientability of E-theory. Hughes, Lau, and I have calculated the ring of functions on the affine scheme $C^k(\mathbb{G}_a, \mathbb{G}_m) \times \mathrm{Spec}(\mathbb{Z}_{(2)})$ , and computational experiment suggests that, further pulling back to $\mathrm{Spec}(\mathbb{F}_2)$ , the Ando-Hopkins-Strickland map $\mathcal{O}_{C^k} \otimes \mathbb{F}_2 \to H_*(BU \langle 2k \rangle; \mathbb{F}_2)$ is very close to being an isomorphism, and the amount it misses it by is exactly the closure of the odd-dimensional parts of the homology under the coaction of the dual Steenrod algebra. This is a bit of a mess of a conjecture, but by hook or by crook we seem to end up with a map from the $(k-1)$ -dimensional analogues of cubical structures to $(2k-4)$ -variate alternating, multiexponential maps.

The real question is: where is this guy in the algebraic geometry literature? The construction used above has no obvious higher dimensional analogue. An equivalent phrasing of the question is: given two line bundles $L_1$ and $L_2$ , we can consider their virtual differences $(1-L_1)$ and $(1 - L_2)$ , which we can put together in creative ways to get a virtual bundle whose classifying map lifts to $BU\langle 6 \rangle$ . These creative ways specifically look rather like the formula for $e_n$ , inexorably leading you toward the Weil pairing story. But, given four line bundles, how do we build a bundle with good properties whose classifying map lifts to $BU\langle 8 \rangle$ ? I, for one, have no idea.

Tags: algebraic geometry, algebraic topology, chromatic homotopy, formal schemes, K-theory
Posted by Eric Peterson in Categories: math | No Comments »

Torus actions and maximal tori, part 2

March 29th, 2011

Way back in part 1, I gave this theorem and promised to sketch a proof:

Theorem 1. If a torus T acts (smoothly) on a compact smooth manifold M with finitely many fixed points, then $|M^T| = \chi(M)$ .

If you’re willing to take this on faith, or just don’t want to think through a proof, you are allowed to skip to the discussion of an even better version of this theorem. But the proof contains some neat ideas, and in the end is just a sketch.

Proof (sketch) of theorem 1. We’ll do this by showing that the Lefschetz number of an appropriate map, homotopic to the identity, is $|M^T|$ . Any torus T has a topological generator, an element t such that the subgroup $\langle t \rangle$ generated by t is dense in T. This means that the fixed points of T are the same as the fixed points of t. The map mentioned in the first sentence is simply the action of t; certainly this is homotopic to the identity since T is connected.

We look separately at the action of T on M near fixed points and away from fixed points. Take U to be a disjoint union of coordinate patches, one around each fixed point, and $V = M \setminus M^T$ . It’s not hard to check that trace is additive, i.e. given a diagram of vector spaces with exact rows

$\begin{array}{ccccc} V_1 & \to & \cdots & \to & V_n\\ {\scriptstyle f_1} \downarrow & & \cdots & & \downarrow {\scriptstyle f_n}\\ V_1 & \to & \cdots & \to & V_n \end{array},$

$\sum_i (-1)^i tr(f_i) = 0$ . Applying this to the Mayer-Vietoris sequence for $U \cup V = M$ shows that if $f : M \to M$ maps each of $U, V, U \cap V$ into itself, then

$L(f) = L(f|_U) + L(f|_V) - L(f|_{U \cap V}).$

$\displaystyle \chi(M) = |M^T| + L(t|_V) - L(t|_{U \cap V}).$

But $t$ has no fixed points on V, so the last two terms are 0 by the Lefschetz fixed point theorem, and we’re done!

Well, not quite. There are two issues that have been glossed over. First, $L(t|_U)$ doesn’t even make sense unless we know t maps U into itself, so a little more care is needed in choosing U. If p is a fixed point for the action any Lie group G, taking differentials yields a linear representation of G on the tangent space $T_p M$ . The relevant fact here is that if G is compact, this also describes the action of G in a neighborhood of p: there is a neighborhood W of p and a G-equivariant diffeomorphism $\phi : W \to T_p M$ with $\phi(p) = 0$ . One can arrange to make these neighborhoods as small as desired, so we can indeed assume that the U chosen above is T-invariant.

Second, the Lefschetz fixed point theorem had some hypotheses on the space, e.g. that it should be a compact smooth manifold, so there is a problem applying it immediately to functions on $U, V, U \cap V$ . But these do deformation retract onto fairly obvious compact submanifolds of M, and the retractions don’t do anything to the fixed points. So that’s cool.

Let’s move on to the promised generalization of Theorem 1.

Theorem 2. If a torus T acts (smoothly) on a compact smooth manifold M, then $M^T$ is a finite disjoint union of embedded submanifolds (perhaps of different dimensions), and $\chi(M^T) = \chi(M)$ .

To see the first assertion about $M^T$ , one uses the result stated above that near a fixed point, the action of T looks like a linear action; since the fixed point set of a linear action is a linear subspace, this gives charts realizing $M^T$ as (a union of) embedded submanifolds. The rest of the proof is analogous to the proof of theorem 1, with appropriate replacements: for example, rather than a neighborhood of a fixed point in which the action of $T$ looks linear, we need a tubular neighborhood of each component of $M^T$ in which the action looks linear.

Here’s an example. A representation $\rho : U(n+1) \to \text{GL}(\mathbb{C}^{k+1})$ of $U(n+1)$ induces an action of $U(n+1)$ on $\mathbb{CP}^k$ , and hence of the maximal torus $D \subseteq U(n+1)$ , consisting of diagonal matrices in $U(n+1)$ , on $\mathbb{CP}^k$ . For example, take V to be the standard representation of $U(2)$ (i.e. $\mathbb{C}^2$ with the usual action), and consider the representation $V \otimes V$ . Picking a basis $x, y$ of V and identifying the corresponding basis $x \otimes x, x \otimes y, y \otimes x, y \otimes y$ with the standard basis of $\mathbb{C}^4$ , the action of D on $\mathbb{CP}^3$ is

$\left(\begin{array}{cc} a & 0 \\ 0 & b \end{array}\right) [z_0, z_1, z_2, z_3] = [a^2 z_0, ab z_1, ab z_2, b^2 z_3]$ .

The fixed points of this D-action are $[1, 0, 0, 0], [0, 0, 0, 1]$ , and $[0, a, b, 0]$ for $[a,b] \in \mathbb{CP}^1$ . So the theorem says $\chi(\mathbb{CP}^3) = 1 + 1 + \chi(\mathbb{CP}^1)$ , and luckily $\chi(\mathbb{CP}^n) = n+1$ , so everything checks out.

More generally, by Schur’s lemma, $\rho|_{D}$ is the sum of 1-dimensional representations $\alpha_1, \ldots, \alpha_{n+1}$ , the weights of $\rho$ . In the example above, if $t = \text{diag}(a,b)$ , then

$\alpha_1(t) = a^2, \alpha_2(t) = \alpha_3(t) = ab, \alpha_4(t) = b^2$ .

The general case will work out just like the example: a weight of multiplicity $m$ will correspond to a submanifold $\mathbb{CP}^{m-1}$ in the fixed point set.

Unfortunately I don’t have any particular cool applications of this theorem to give, like the theorem on maximal tori last time. If nothing else, let’s show that any homogeneous space for a compact connected Lie group has nonnegative Euler characteristic; that is, if G is connected and compact, and H is a closed subgroup, then $\chi(G/H) \geq 0$ . However, we’ll only need Theorem 1.

Fix a maximal torus T in G, and consider the left action of T on G/H. It’s easy to write down the fixed points for this action. Let $K = \{g \in G : g^{-1} T g \subseteq H\}$ ; then the fixed point set is $(G/H)^T = K/H$ (this is perhaps slightly misleading notation, since K need not be a subgroup of G). The rank of a compact Lie group is the dimension of the largest torus subgroup: if $\text{rank}(H) < \text{rank}(G)$ , then we see that $(G/H)^T$ is empty. In this case, Theorem 1 implies $\chi(G/H) = 0$ .

So, suppose $\text{rank}(G) = \text{rank}(H)$ . We can assume that $T \subseteq H$ . By the theorem we proved in part 1, any two maximal tori in H are conjugate by a member of H: thus, if $g \in K$ , then there is $h \in H$ with $g^{-1} T g = h^{-1} T h$ . But this shows that $gh^{-1}$ normalizes T, so we conclude that $K = N_G(T) H$ , where $N_G(T)$ is the normalizer of T in G. Now

$K/H = (N_G(T)H) / H \simeq N_G(T) / (H \cap N_G(T))$

(these are not groups, but they’re still in bijection). Since $T \subseteq H \cap N_G(T)$ , we see that $K/H$ is naturally embedded in $N_G(T) / T$ . In the previous post I mentioned that $N_G(T) / T$ is finite (it’s the Weyl group of G), and so we see that $K/H$ is also finite (and nonempty). Therefore $\chi(G/H) = \chi(K/H) > 0$ . In fact, we get a stronger result than initially stated:

Theorem. If $\text{rank}(H) < \text{rank}(G)$ , then $\chi(G/H) = 0$ . If $\text{rank}(H) = \text{rank}(G)$ , and $W$ is the Weyl group of $G$ , then $\chi(G/H)$ is a positive integer divisor of $|W|$ .

Tags: algebraic topology, lie theory, representation theory
Posted by Brendan Pawlowski in Categories: math | No Comments »

Spectral sequences on one blackboard

March 10th, 2011

Spectral sequences are very often referred to as the most forbidding part of learning homological algebra or algebraic topology. The standard presentation is an absolute sea of indices, connecting homomorphisms, kernels and images, filtrations and associated gradeds. They appear only when the lecturer thinks that there’s no way around them, and frequently the lecturer has already made a couple of arguments in the “style” of spectral sequences without actually using those words in some unhappy effort to familiarize the student with the sea of indicies that’s about to come.

But I’m here to tell you that spectral sequences don’t have to be this way! Spectral sequences fit on one board, using one picture, with one index, with obvious motivation, and with minimal misery. This presentation is of course not at all new. It’s still worth saying again in public, on record.

To begin, the key property of a homology functor is that it sends short exact sequences (or cofiber sequences or distinguished triangles or whatever your favorite phrasing and context are) to long exact sequences of homology groups. This is useful in what are now considered very standard ways: objects can be built up out of smaller ones through extensions of these types, and then the homology of the new object is related to the homologies of the old ones.

But, it is rarely the case that objects are given to us in one-step constructions! It is commonly the case that we have a whole filtration: a sequence of subobjects $F_i$ , each one stacked in the next, like so:

Then for each of these inclusions $F_{i+1} \to F_i$ , we can build a quotient / cokernel / cofiber object, like this:

Each of these angled pair of arrows is a cofiber sequence, so when we apply homology to this entire diagram, we get going-around maps from the snake lemma:

These bottom groups are called the first page of the spectral sequence, notated $E^1_{p, q} = H_p F_q / F_{q+1}$ (though this grading varies by author and application). The spectral sequence is going to be an object which interrelates all these long exact sequences.

This diagram as drawn is no longer commutative, so you’ll have to keep aware of that. Instead, each of these triangles is exact in the usual sense that at each node, the image of one map is equal to the kernel of the next. Now, keep in mind our goal: we’re supposed to be reconstructing $H_* A$ from these filtration quotients $H_* F_i / F_{i+1}$ . There are two clear problems with looking at the homology of the filtration quotients:

Maybe we have a cycle that ought to be the boundary of something, but we’re missing that boundary class in this filtration level, or
Maybe we have an extra cycle coming from collapsing the lower filtration quotient, where there ought to be an obstruction to building the cycle.

These two things are kept track of by arrows that we’ve actually already drawn. Namely, if we choose a class $x \in H_* F_i / F_{i+1}$ that ‘deserves’ to be the preimage of a boundary for some cycle in $H_* F_{i+1} / F_{i+2}$ , then the composition $\pi_{i+1} \circ \partial_i$ will spit out that class when applied to $x$ . So, we define a sequence of maps $d_1 = \pi_{*+1} \circ \partial_*: H_* F_i / F_{i+1} \to F_{i+1} / F_{i+2}$ .

These maps in fact form a chain complex — namely, $d_1 \circ d_1 = \pi_{*+2} \circ \partial_{*+1} \circ \pi_{*+1} \circ \partial_*$ , and in the middle of this composition we have $\partial_{*+1} \circ \pi_{*+1} = 0$ , since they’re two edges of the same exact triangle. So, it makes sense to take homology with respect to these maps, doing which will begin to sort out this business with extra cycles lying around.

So, suppose we do that. The resulting groups $E^2_{p, q}$ are called the second page of the spectral sequence — but then what? Well, what’s left in any filtration degree now is a subquotient of what we started with: we selected only the cycles which did not deserve to be called boundaries one level up and which were not themselves bounded by things one level down — so, in particular, for a class $y$ in these new groups, we have $d_1 y = 0$ . So, what does that mean? The map $d_1$ is defined by $\pi_{*+1} \circ \partial_*$ , and so we have that $\partial_* y$ is in the kernel of $\pi_{*+1}$ . This, in turn, means that $\partial_* y$ has a well-defined preimage $i^{-1}_{*+1} \partial_* y$ , which we can again push down to $\pi_{*+2} i^{-1}_{*+1} \partial_* y = d_2$ . These are sketched as the dashed maps below:

Once again, these maps check whether $y$ ought to be involved with the boundary classes in a higher filtration degree — this time two levels up. Once again, the maps $d_2$ form a chain complex and once again taking their homology helps sort out where boundaries belong and don’t belong. This process continues forever, on to higher and higher order pages.

It’s believable, on the face of it, that these groups ‘limit’ to something meaningful with respect to the homology $H_* A$ (called the abutment), but there are fairly serious technical conditions here governing what actually happens (for instance, Boardman wrote a famous 40-ish page paper about them titled Conditionally Convergent Spectral Sequences). The most basic condition to request is that for each homological degree $q$ , there is a filtration degree $p$ such that for $p' > p$ , $H_q F_{p'} = 0$ . This forces these groups to stabilize because it forces us to eventually run out of differentials for each group in the spectral sequence. Then, provided these limiting groups make sense, what’s left over on the $\infty$ -page corresponds to filtration quotients of the homology $H_* A$ — so we’ve traded the homology of the filtration for the filtration of the homology. This new problem of stitching the groups together through extensions to recover $H_* A$ is nontrivial but at least tractable — it’s an algebraic question about homology groups rather than a question about whatever source objects we started with.

This is frequently satisfied. For instance, let $X$ be a CW-complex of dimension $n$ , and let the filtration $F_k = X^{(n-k)}$ be the skeletal filtration. Then $F_k / F_{k+1}$ is a wedge of spheres, whose singular homology is easily understandable, and the information in the associated spectral sequence is concentrated in such a way that only the differential $d_1$ could possibly be nonzero; all the others vanish for degree reasons. These differentials $d_1$ actually compute the cellular homology of the space, and again the degrees of the nonvanishing groups are stratified in such a way that there are no nontrivial extension problems either. This spectral sequence is said to collapse at the second page, and this proves that cellular homology computes singular homology, just like that. Similarly, we could have used an ascending filtration rather than a descending one, which would remove the dimension restriction on our CW complexes, or we could have used cohomology instead of homology to get a dual statement. More interestingly, we didn’t have to use singular homology at all — we can replace this with any extraordinary homology theory to produce the Atiyah-Hirzebruch spectral sequence, beginning with cellular homology with coefficients in the coefficient ring of some bizarre homology theory and converging to the extraordinary homology of the space itself. Or, starting with a fiber bundle over a CW-complex, we can filter the total space by preimages of the skeleta in the base, which (though it takes some work to show this) produces the Serre spectral sequence.

Here’s a more radical example of how you can easily roll your own spectral sequences with this setup: one model for the classifying space $BG$ uses a simplicial version of the bar construction on $G$ . Filtering the resulting simplicial complex by dimension, we produce a spectral sequence whose first page looks like $E_* G$ , $E_* G^{\otimes 2}$ , $E_* G^{\otimes 3}$ , … . And, one checks, using methods entirely similar to the cellular homology spectral sequence, that the differential $d_1$ is the differential for the bar complex on the algebra $E_* G$ , and hence the second page of the spectral sequence begins with $\mathrm{Tor}^{E_* G}_{*, *}(E_*, E_*)$ , converging to $E_* BG$ . In this sense, those Tor groups (sometimes referred to as the algebra-homology of $E_* G$ ) are a first-order approximation to the homology of the classifying space. The spectral sequence thus intertwines taking $E_*$ -homology and building the bar complex.

Of course, you don’t have to get fancy at all to see this construction work. You could try the toy example filtration $\mathrm{pt} \to S^1 \to D^2$ of $D^2$ , a contractible space. The two interesting filtration quotients are $\mathrm{pt} \to S^1 \to S^1$ and $S^1 \to D^2 \to S^2$ , exemplifying the two cases above where we might have accidental extra cycles. The spectral sequence for the homology of these spaces therefore contains two nonzero groups, and these knock each other out when moving to the second page — which is then completely empty, just like we wanted.

Tags: homological algebra, spectral sequences
Posted by Eric Peterson in Categories: math | 3 Comments »

Motivating Goodwillie’s construction

February 24th, 2011

I have tried at several points over the past year or so to get through Goodwillie’s three papers on his calculus of functors, but each time I make it through just a handful of pages before getting discouraged or distracted. Some parts of it have started to sink in, though, and I wanted to share what I’ve collected so far with you all. Without a doubt this viewpoint is known to everyone versed in the subject, but I haven’t found this analogy written down publicly yet. Here goes:

Classical Taylor polynomials: Polynomial functions are a nice class of functions to consider for us because they have two important properties: first, they have an ascending filtration by degree, so that if a polynomial is of degree at most n then it also is of degree at most n+1, and second, a degree n polynomial is completely determined by its value at any (n+1) sample points. Given an arbitrary function f, we can build a polynomial of degree n $P_n f$ , called a Taylor polynomial, which is sort of the best polynomial approximation to f at a fixed basepoint (say 0) in the following sense: $P_n P_{n+1} f = P_n f$ and $P_{n+1} P_n f = P_n f$ . These polynomials can be constructed in a variety of ways, but here’s a cool one that’s relevant for us:

Pick (n+1) sample points $x_0, \ldots, x_n$ , and build the interpolating polynomial $T_n f$ passing through the points $(x_i, f(x_i))$ . Then, $P_n f$ is defined using the limit

$P_n f = \lim_{x_0, \ldots, x_n \to 0} T_n f.$

If you haven’t seen this before, then it’s a good idea to work through an example. Say $f(x) = e^x$ , $n = 2$ , and $(x_0, x_1, x_2) = (0, h, -h)$ . Then $T_2 f = ax^2 + bx + c$ , where you use whatever your favorite computational method is to determine that $c = 1$ , $b = \frac{e^h - e^{-h}}{2h}$ , and $a = \frac{e^{-h}-2+e^h}{2h^2}$ . Using L’Hopital and to take limits, we find that $c \to 1$ , $b \to 1$ , and $c \to \frac{1}{2}$ , and hence $P_2 \exp = 1 + x + \frac{1}{2} x^2$ . I promise that this works in general; pick an arbitrary function and number of points, and you’ll get the first chunk of its Maclaurin expansion.

Constructing the functors $P_n F$ :

We’ll now spend the rest of this discussion translating this setup, line by line, into the ridiculous context of homotopy functors $F: C \to D$ between pointed simplicial model categories C and D.¹ The first thing we should do is specify what it means to pick a collection of sample points and their relation to a favorite basepoint; to do so, we’ll introduce cubes. Let [d] denote the set $\{1, \ldots, d\}$ . Then, $P[d]$ , the power set of [d], is a partially ordered set and hence a category. A d-cube in C, then, is a diagram $X: P[d] \to C$ ; the reason for the name is that the indexing category P[1] looks like a line, P[2] like a square, P[3] like a cube, and so on through the hypercubes. A cube is said to be Cartesian (resp. coCartesian) if the initial (resp. final) vertex is weakly equivalent to the homotopy limit (resp. colimit) of the remainder of the cube with that vertex deleted. A cube being (co)Cartesian is a statement about redundancy; the data contained in these initial and terminal corners are recoverable using just the rest of the cube and a limiting or gluing procedure. There’s also a more extreme form of redundancy: a d-cube is said to be strongly coCartesian if all of its faces of dimension at least 2 are themselves coCartesian. When d is at least 2, this clearly implies that the cube is itself coCartesian; we’re asserting that to specify a strongly coCartesian cube, you need to describe the initial vertex, d maps of the form $X(\emptyset) \to X(\{i\})$ for $1 \le i \le d$ , and everything else can be fleshed out by taking homotopy pushouts from there.

Now, we can define our analogues of polynomials: F is said to be d-excisive (or of degree at most d) when it sends strongly coCartesian (d+1)-cubes to Cartesian (d+1)-cubes. This completes the analogy of the first paragraph of this post: a d-excisive functor is also (d+1)-excisive, so these classes of functors is ascending, and this business about being strongly coCartesian corresponds to selecting an interesting basepoint value $X(\emptyset)$ , (d+1) sample points together with their relations to the basepoint, and then after applying F just to the extraneous sample points we can reconstruct $F(X(\emptyset))$ by taking a homotopy limit.²

In our example, I didn’t select arbitrary sample points $(x_0, x_1, x_2)$ , but instead I picked $(0, h, -h)$ to ease our computation. We’re going to do something similar in the Goodwillie setting; for a finite set T and an object $A \in C$ , define $A \ast T = \mathrm{hocofib}\left(\bigvee_T A \to A\right)$ , so $A*1$ is the cone on A, $A*2$ is the suspension or a 2-pointed cone, $A*3$ is a 3-pointed cone, and so on. For a fixed A and $T \in P[d+1]$ , the assignment $A \mapsto A * T$ is a strongly coCartesian cube, which is going to be our analogue of picking smart sample points.³ To compute $T_d F(A)$ , we basically force excisiveness to hold:

$T_d F(A) = \mathrm{holim}_{T \in P[d+1] \setminus \{\emptyset\}} F(A \ast T).$

Since $F(A)$ itself gives a top corner for the cube and $T_d F(A)$ is defined with a limit, we have a natural map $F(A) \to T_d F(A)$ . Our analogue of letting the sample points tend to the basepoint doesn’t make much sense, but we do something that looks vaguely analogous:

$P_d F(A) = \mathrm{hocolim}(F(A) \to T_d F(A) \to T_d T_d F(A) \to \cdots).$

It is pretty difficult to show that this is the right construction — in fact, Goodwillie writes in his paper that his own proof hardly makes sense to him. The key point is that the map $F(A * T) \to T_d F(A * T)$ factors through a Cartesian (d+1)-cube Y, i.e., we have maps $F(A * T) \to Y \to T_d F(A * T)$ . At each stage of the sequential limit defining $P_d F$ , then, we can insert one of these Cartesian cubes and instead take a homotopy colimit through them, which guarantees that what we get in the end will be d-excisive.⁴

Properties of the Taylor functors:

There’s a natural map $e_d: F \to P_d F$ . Because the natural map $F \to T_d F$ is an equivalence when F is d-excisive, $e_d$ is also an equivalence when F is d-excisive. So, because $P_d F$ is d-excisive, we get $e_{d+1}: P_d F \to P_{d+1} P_d F$ is a weak equivalence, which is one of the special properties of the original Taylor polynomials. The other equivalence, $P_d F \simeq P_d P_{d+1} F$ , is a little more technical; it requires results about sequential homotopy colimits commuting with finite homotopy limits, but I promise that it’s true too. This second equivalence gives us a map $p_d: P_{d+1} F \to P_d P_{d+1} F \simeq P_d F$ , and hence these approximating functors assemble into one big tower. A functor is said to be analytic when it’s weakly equivalent to the limit of this tower, or with a radius of convergence when it’s sometimes weakly equivalent to the limit of the tower.

The real utility of these properties is that they imply the universality of Goodwillie’s construction. If $\eta: F \to G$ is a map to a d-excisive functor, then there exists a zigzag $P_d F \xleftarrow{\simeq} G' \to G$ factoring $\eta$ .

Taylor’s theorem:

When you read the example at the top of the page with the second order expansion of the exponential, you believed that what I was saying was true because you recognized the series — you already knew how to compute it. Namely, Taylor’s theorem gives us an express description of these polynomials as $P_d f = \sum_{n=0}^d \frac{f^{(n)}(0) x^n}{n!}$ . This formula, miraculously, is mirrored in the Goodwillie calculus (for stable model categories). I will omit absolutely all details; I just want to give an example of the depth of the analogy we’re pursuing.

Define $D_d f = P_d f - P_{d-1} f = \frac{f^{(d)}(0) x^d}{d!}$ to be the part of f which is homogeneous of degree d. On the level of functors, we have $D_d F(A) = \mathrm{hofib} P_d F(A) \xrightarrow{p_d} P_{d-1} F(A)$ . This object gives us one way of measuring when a functor is (d-1)-excisive; if it were, then $D_d F$ would vanish everywhere. There’s another way to test for excisiveness called cross-effects:

$cr_d F(X_1, \ldots, X_n) = \mathrm{hofib}(F(X_1 \vee \cdots \vee X_n) \to \mathrm{holim} F(\hbox{cube of wedges})),$

where the cube of wedges is the strongly coCartesian cube whose T^th vertex is of the form $\bigvee_{t \in T} X_t$ . The cross-effects will also vanish when F is d-excisive, and it’s natural to ask how these two methods compare. Goodwillie produces the following formulas:

$D^{(d)} F = cr_d D_d F,$
$\partial^{(d)} F = D^{(d)} F(\mathbb{S}, \ldots, \mathbb{S}),$
$D_d F(X) = (\partial^{(d)} F \wedge X^{\wedge d})_{h \Sigma_d}.$

This is really a magical thing to write. This spectrum $\partial^{(d)} F$ produced by applying the cross effects functor to the sphere spectrum plays the role of the d^th derivative $f^{(d)}(0)$ , and then we also smash against d copies of X, just as in Taylor’s formula. Finally, the analogy for dividing by $d!$ is the homotopy quotient by the evident action of the symmetric group on d letters on this smash product. Cute!

(Note: This is maybe not the best presentation of the Taylor coefficient, since it requires you to already know $D_d$ to compute it, which is sort of contrary to the point. There are other ways to compute the coefficient by linearizing the cross effects functor, without applying $D_d$ at any point.)

If you’re interested in reading more, you should pull up Goodwillie’s three papers, amusingly titled Calculus 1, Calculus 2, and Calculus 3, or you can check out Kuhn’s smartly-written, example-rich introduction to the calculus in general and also its interactions with and applications to chromatic homotopy theory.

—

¹ – If you don’t have the stomach for model categories, you can always specialize to the case of spaces or, even better, spectra. Mostly I just want to be able to take homotopy (co)limits.
² – To check your understanding of the definition, you should check that homology functors are 1-excisive, or linear. In fact, Brown representability says that 1-excisive functors taking coproducts to coproducts are all homology functors. Without this second condition, 1-excisive functors are harder to classify; on spectra, mapping space functors and Bousfield localizations are 1-excisive, for instance!
³ – This is maybe more like picking a primitive (n+1)^th root of unity $\zeta$ and then using $x_i = \zeta^i h$ as the interpolating points, rather than using the points I picked above.
⁴ – To check that you understand this construction, let’s compute $P_1 \mathrm{id}_{\mathsf{Spaces}}$ . We pick an object A, then take a homotopy colimit along $CA \simeq T * \{0\} \leftarrow A \to T * \{1\} \simeq CA$ to fill the square out with the suspension of A. Then, we apply the identity functor to the corner $CA \to \Sigma A \leftarrow CA$ and take a homotopy pullback of the weakly equivalent corner $P \Sigma A \to \Sigma A \leftarrow P \Sigma A$ to compute $T_1 \mathrm{id}(A) = \Omega \Sigma A$ , and hence $P_1 \mathrm{id}(A) = \Omega^\infty \Sigma^\infty A = Q A$ !

Tags: algebraic topology, homotopical algebra, magic, talks
Posted by Eric Peterson in Categories: math | No Comments »

Minimal Models, Maximal Mathematics: Part I

February 17th, 2011

So, we’ve got a topological space, huh? Let’s call it $X$ . What are we gonna do with it? Well, the first thing that we might try to do is compute it’s homotopy groups, $\pi_n (X)$ . That seems like a pretty normal thing to do, being topologists. Unfortunately for us, computing homotopy groups is (generally) a Pretty Hard Thing. We might try and get around some of this difficulty by only asking for the rational homotopy groups of $X$ , which we obtain by taking $\pi_q (X) \otimes \mathbb{Q}$ . This effectively kills the torsion in $\pi_q (X)$ , and shows us only the rank of $\pi_q (X)$ . We’ve thrown away some of the information contained in $\pi_q (X)$ by doing this, but the end result is that we get something that can often be computed.

How, you ask? Well, one way that this can be done is via the theory of minimal models, which was introduced by Sullivan in the 70′s. We’re getting paid by the theorem, so let’s get to it! As usual, there’s a bit of setup involved in starting to talk about this business, but it’ll pay off in the end.

In the following, all algebras are over $\mathbb{R}.$ Let $A = \oplus_{i \geq 0} A^i$ be a differential graded commutative algebra, with the differential $d$ being an antiderivation of degree 1 (it moves things up a degree):

$d(a \cdot b ) = (da) \cdot b + (-1)^{|a|} a \cdot db$

with graded commutativity:

$a \cdot b = (-1)^{|a||b|} b \cdot a$ .

An example of this is the De Rham complex of differential forms on a smooth manifold, with the product being the wedge product and the differential being the exterior derivative.

Definition. We say that such an algebra is free if it satisfies no relations besides associativity and graded commutativity. We write $\Lambda (x_1, \dots, x_n)$ for the free algebra generated by $x_1, \dots, x_n.$ An element in $A$ is said to be decomposable if it is a sum of products of positive elements in $A$ (degree > 0).

Now, we introduce minimal models, which will be our main objects of study.

Definition. A differential graded algebra $M$ is a minimal model for $A$ if:

$M$ is free;
there is a chain map $f : M \rightarrow A$ that induces an isomorphism on cohomology;
the differential of a generator is either zero or decomposable.

Definition. A differential graded algebra $A$ is 1-connected if $H^0 (A)= \mathbb{R}$ and $H^1 (A) = 0$ .

Note that the De Rham cohomology of any connected, simply-connected manifold will have this property. The nice thing about such 1-connected differential algebras is that they have minimal models. That is:

Proposition. Let $A$ be a graded, differential $\mathbb{R}$ -algebra that is 1-connected. Then A has a minimal model.

The proof is easy enough, and you’d probably get the idea from the fact that they’re called “minimal models”. We’re going to construct the minimal model $M$ for $A$ by taking free algebras and adding in exactly the extra stuff that we need in order for $M$ to have the same cohomology as $A$ (hence the “minimal”).

Let’s start off trying to get $H^2(M)$ right. That’s not too hard: take $a_1, \dots, a_n$ to be generators of $H^2 (A)$ , and define $M_2 = \Lambda (a_1, \dots, a_n)$ , setting $|a_i| = 2$ and $d a_i = 0$ . The map from $M_2$ to $A$ is given by

$f: M_2 \rightarrow A$

$a_i \mapsto a_i$ .

As of now, $f$ induces an isomorphism on cohomology (there are no elements in $H^1 (A)$ by hypothesis, and the generators of $H^2 (A)$ are all represented). Moreover, $f$ is an injection in degree 3, as there are no elements in $M_2$ of degree 3. This is going to be the base for an inductive construction of the minimal model. That is, we’ll show that for any $n$ there is a minimal free algebra $M_n$ and a chain map $f : M_n \rightarrow A$ so that

the algebra $M_n$ has no elements in degree 1 and no generators in any degree greater than $n$
the map $f$ induces an isomorphism in cohomology in dimensions less than $n + 1$ and an injection in dimension $n +1$ .

Suppose that this is true for $n = q-1$ . By assumption, we’ve got exact sequences

$0 \rightarrow H^q (M_{q-1}) \rightarrow H^q (A) \rightarrow \mathrm{coker} \, H^q (f) \rightarrow 0$

and

$0 \rightarrow \mathrm{ker} \, H^{q+1} (f) \rightarrow H^{q+1} (M_{q+1}) \rightarrow H^{q+1} (A).$

The first comes from the hypothesis that $f$ is an injection in degree $q$ , and the second is simply a rephrasing of what it means to be in $\mathrm{ker} \, H^{q+1} (f).$ The idea is to introduce elements into $M_{q-1}$ in order to kill $\mathrm{coker} \, H^q (f)$ and $\mathrm{ker} \, H^{q+1} (f)$ . If we do this, the above short exact sequences give us an isomorphism and injection, respectively (which is exactly what we want for the induction).

Let $\{ [b_i] \}$ be a basis of $\mathrm{coker} \, H^q (f)$ , and

Define

$M_q = M_{q-1} \otimes \Lambda (b_i, \xi_j), \; \; \dim b_i = \dim \xi_j = q.$

$M_q$ is a free minimal algebra, if we define the differential

$d(m \otimes 1 ) = ( dm ) \otimes 1 ,$

$d(1 \otimes b_i) = 0,$

$d(1 \otimes \xi_j) = x_j \otimes 1.$

We extend $f$ to $M_q$ by

$f(m \otimes 1 ) = f(m)$

$f(1 \otimes b_i) = b_i$

$f(1 \otimes xi_j) = \alpha_j$

where $\alpha_j$ is an element of $A$ such that $f(x_j) = d\alpha_j$ (which exists, by $x_j$ being a basis of $\mathrm{ker} H^{q+1}(f)$

Checking that this new $f$ satisfies the properties required in the induction step is a simple exercise, and is pretty believable after a moment’s thought (we’ve killed the appropriate elements in the short exact sequences written earlier). Because the cohomology of $A$ is assumed to be finite dimensional, the $f$ ‘s constructed become isomorphisms after the stage corresponding to the maximal dimension in $H^{*} (A)$ is reached, giving us a minimal model for $A$ .

So, we’ve got ourselves some minimal models. What do we do with them? It seems like we need to know the cohomology of $X$ in order to build the minimal model, so one might wonder if we actually get any information out of it. As promised earlier, though, we actually get some homotopy-theoretic information out of a minimal model, as shown by the following theorem of Sullivan:

Theorem. Let $X$ be a simply connected manifold and $M$ its minimal model. Then the dimension of the vector space $\pi_q (M) \otimes \mathbb{Q}$ is the number of generators of the minimal model $M$ in dimension $q$ .

That’s great (even though it’s actually a bit difficult to prove – see Sullivan’s paper Infinitesimal Computations in Topology for a proof and a huge amount of other stuff)! It turns out that the minimal model lets us detect the non-torsion parts of the homotopy groups, and in a totally computable fashion. Let’s do an example to illustrate the usefulness of this setup.

Proposition. The homotopy groups of an odd sphere $S^{2n+1}$ are torsion except in dimension $2n+1$ , where it is infinite cyclic; for an even sphere $S^{2n}$ , the exceptional dimensions are $2n$ and $4n -1$ .

We’ll show this by simply exhibiting minimal models for spheres. Let’s start with an odd-dimensional sphere. We have that the De Rham cohomology of $S^{2n+1}$ is the exterior algebra on one generator of degree $2n+1$ (corresponding to the volume form of $S^{2n+1}$ . So, we have that the cohomology is its own minimal model. By the above theorem, the only homotopy groups of $S^{2n+1}$ with free part are $\pi_{2n+1}$ .

For an even-dimensional sphere the situation is slightly more complicated – but not much more. The De Rham cohomology of the sphere $S^{2n}$ is not an exterior algebra because we do not have that anti-symmetry kills the element corresponding to the volume form. That is, if $\omega$ is the volume form, we need to specify that $\omega \wedge \omega = 0$ – this is not forced by the exterior algebra structure. We start off with $\Lambda(x)$ , the exterior algebra on one generator $x$ with the degree of $x$ being $2n$ (corresponding to the volume form of the sphere) and set $dx = 0$ . We need that $x^2$ (corresponding to $\omega \wedge \omega$ ) be 0 in the cohomology of the minimal model, so we stick in another generator $y$ (in degree $4n -1$ , as $x^2$ has degree $4n$ and $d$ moves us up in degree by 1) and define the differential so that

$dy = x^2$

This kills $x^2$ in cohomology, and because the degree of $y$ is odd, $y^2$ is 0 automatically and we do not need to add in any more generators. This gives us a minimal model for the cohomology of the even sphere, and the map from this model to the De Rham complex of the sphere is given by

$x \mapsto$ volume form of the sphere

$y \mapsto 0$

Again, by the above theorem, we have that the only homotopy groups with positive rank are $\pi_{2n}$ and $\pi_{4n-1}$ . Fancy. A good exercise is to do the same computation for the complex projective spaces $\mathbb{CP}^n$ – it’s basically the same deal as with the sphere.

We can get even crazier though, even though I’m not going to do it here. The wedge of the spheres $S^n$ and $S^m$ is the union of $S^n$ and $S^m$ with one point in common, written $S^n \vee S^m$ . Using the above theorem, it’s not too tough (with a bit of knowledge of differential topology) to compute some of the rational homotopy groups of, say, $S^2 \vee S^2$ — something that would be difficult to approach otherwise.

Computing rational homotopy groups is not all that you can do with this setup, however! It turns out (by another theorem of Sullivan) that if we have a suitably nice manifold $X$ , any homomorphism from $\pi_n(X) \rightarrow \mathbb{R}$ is realized by integrating the pullbacks of some appropriately chosen differential forms and their primitives. We’ll discuss this more next time!

Tags: algebraic topology, dgas, rational homotopy
Posted by Matthew Pancia in Categories: math | 1 Comment »

Links
- arXiv front for math
Meta
- Log in
- Valid XHTML
- XFN
- WordPress

Chromotopy

Research and bros

Higher order Weil pairings

Torus actions and maximal tori, part 2

Spectral sequences on one blackboard

Motivating Goodwillie’s construction

Minimal Models, Maximal Mathematics: Part I

Tags

Authors

Archives

Links

Meta