Reading Responses

In this section, Professor Haynes Miller describes the reading responses students complete for each paper they do not present.

Each participant submits a “reading response” to each paper he or she is not presenting, in advance of the talk, either on paper or by email. I reply by email to each reading response. Initially, students are often unsure of what is expected of them. I explain that I don’t want them to provide me with a sequence of statements and theorems; rather, I want to hear what they’re reminded of by this paper, what connections they see, or what surprised them. It’s an opportunity for them to engage me in a discussion about those connections and revelations.

Although they all produce deep and engaging responses, they are all so different! This is what makes responding to their writing so interesting for me as an instructor. Sometimes students pose questions in their responses. I usually cannot answer their specific questions, but I try to point them to resources and references that will help them. I almost always use their questions as springboards for writing about a related topic. Quite often we engage in a brief “back and forth” email conversation. I think it’s fantastic—It means students are really thinking and learning.

Sample Correspondence

Below, Professor Haynes Miller and his student, Isabel Vogt, share their first reading response exchange.

4 Sep

Response to: Steenrod Operations (PDF - 1.7MB)

Dear Professor Miller,

Here is my response:

Before reading your notes I had seen the definition of the Steenrod squares given in Mosher-Tangora “Cohomology Operations and Applications in Homotopy Theory”. This approach makes more concerted use of the cell structure of B_(Z/2Z) = RP^\infty and E_(Z/2Z) = S^\infty and defines the Sq^i in terms of cup_i products on integral cochains.

I think that I understand the more geometric construction based upon the pi-adic construction on X that you present better than this construction, but I still have very little intuition for these operators geometrically based upon the definition. I was hoping you could help me understand this.

Supposed X is a smooth manifold. Then any cohomology class in H^q(X) corresponds under Poincare duality to a linear combination of codim q submanifolds. For u_Y a class in H^q(X) corresponding to a cidim q sub manifold Y, you can compute u_Y \cup u_Y as the cohomology class of the self-intersection of this submanifold — e.g. as the cohomology class of the zero locus of a section of the N_(Y/X) (or as the index q Stiefel-Whitney class of the normal bundle).

I’ve heard that Sq^i(u_Y) corresponds to the index i Stiefel-Whitney class of Y. This gives me more intuition for these cohomology operations.

Is there a way to understand this in terms of the construction you gave?

Thanks!
Isabel

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7 Sep

Dear Isabel,

A more modern treatment of the cup-i approach is contained in the statement that the cochain functor (to DG algebras) lifts to a functor to E_\infty algebras. See Mandell, Cochain Multiplications, Am J Math 2002. This approach has its own virtues; for example, the cohomology of any co-commutative Hopf algebra comes equipped with Steenrod operations. (But they don’t satisfy Sq^0=1; instead, they are induced by the Frobenius in the dual Hopf algebra.)

I think a reference for the construction you describe is McCrory, Cobordism operations and singularities of maps, BAMS 82 (1976) 281ff. I guess it’s related to the statement, which I think we will hear from [Student Name], that SW classes correspond under the Thom isomorphism with Steenrod operations on the Thom class.

Haynes

The complete correspondence between Professor Miller and Isabel is available in the table.

MONTHS	SAMPLE CORRESPONDENCE
September	Sample Correspondence
October	Sample Correspondence
November	Sample Correspondence
December	Sample Correspondence

3 Dec

Response to: Tillmann, U. “Mumford’s Conjecture—A Topological Outlook.” In Handbook of Moduli: Volume III (Advanced Lectures in Mathematics). Vol. 26. International Press of Boston, 2013, pp. 399–429. ISBN: 9781571462596.

Dear Professor Miller,

Here is my response to the Tillmann Paper:

Is the idea that M_g^top is easier to work with since its a classifying space, but for proving Mumford;s conjecture, the fact that it is rationally equivalent to M_g is enough? It seems that M_g^top is interesting in its own right, especially given the homotopy theory in the rest of the paper.
Is there any geometric intuition for Mumford’s conjecture? It seems that in the way it is phrased it is very geometric, but the proof relies on topological techniques.
What is the main take-away from this? How does knowing this result that the stable rational cohomology of M_g is a polynomial algebra on some explicit generators prove useful? Or what is the take-away from the broader topological framework?

Thanks!
Isabel

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3 Dec

Isabel

I agree. Coarse moduli spaces are just weird.
Well, the classes \kappa_i are very natural, and nobody could think of any other way to generate characteristic classes for surface bundles. That’s when conjectures bubble up. What Mumford did not conjecture, because it requires topological techniques, is that the scanning map is an integral equivalence, not just rational.

The question of the relations among those classes in H^*(BDiff_g) is very interesting. I think the image is actually finite dimensional.
This was a big step forward in understanding characteristic classes for surface bundles. Perhaps it’s unfair to ask why one would be interested in such things in the first place. The techniques have proven very important too, and have led to a whole range of analogous theorems about diffeomorphism groups in higher and lower dimensions. In lower dim’s, you are looking at homotopy automorphisms of graphs, or what is the same the automorphism group of a free group. These can again be stabilized, and Soren Galatius worked out what the analogous infinite loop space is. In higher dimensions we don’t have such a clean way to stabilize, but there are still results around (by Galatius and others).

Also the spectrum MTSO(d) has been seen before. An alias is Th(-can | BSO(d)) where can is the canonical d-plane bundle and by Th I mean the Thom spectrum of the virtual vector bundle associated to -can.

Haynes

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5 Dec

Response to: Adams, J. F. “Quillen’s Work on Formal Groups and Complex Cobordism.” In Stable Homotopy and Generalised Homology (Chicago Lectures in Mathematics). University of Chicago Press, 1995. ISBN: 9780226005249. [reprint of the 1974 original]

Hi Professor Miller,

My response is below:

First of all, reading this paper was really satisfying! I’ve really only seen the usefulness of formal groups from the angle of number theory, so I enjoyed this a lot, and I’m glad I finally read it.

More particular things:

The Lazard ring really measures the constraints put on a formal group to be associative and commutative, right? But then this intuitive definition (which obviously satisfies the desired universal property) doesn’t give a satisfactory tool because of the relations. So Lazard’s theorem can be thought of as giving something more of a practical handle on this?
You’re right, given Milnor’s result on pi_*(MU) being a polynomial algebra in generators of the right degree, Quillen’s theorem just feels like it has to be true for algebraic reasons! Is there any intuition? Also, I understand what the Lazard ring means… is there some reason why I should understand pi_*(MU) to satisfy the same universal properties? Is this isomorphism canonical (I think no? Was there choice involved in the formal group law?)?
Cor 9.2 of section 9 makes me think immediately of multiplicative genera….

Thanks!
Isabel

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5 Dec

Isabel

Lazard’s theorem is equivalent to the statement that any (1d, commutative) formal group can be lifted across any surjection of rings. The annoying thing about Lazard’s theorem is that it doesn’t provide you with generators. This defect is one of the reasons for liking the condition of p-typicality.

Quillen’s approach is to try to show that the generators of the MU formal group law generate MU_*. He does this using a version of Steenrod operations on MU_*.

Suppose you have a commutative monoid R in the stable homotopy category (a “ring spectrum”) whose homotopy groups are zero in odd dimensions (eg K-theory). Then you can compute that R^*(CP^\infty) is a power series ring over R^*, and you get a formal group over R^*. The set of coordinates for this formal group are in bijection with the set of ring spectrum maps from MU.

Formal groups provide an interpretation genera for U-manfolds. All the parts of the theory of genera have interpretations in terms of formal groups. The characteristic series is x/exp(x), for example. I think Miscenko’s theorem is exceptionally beautiful!

Haynes

5 Nov

Response to: Quillen, D. G. “Algebraic K–theory I.” In Higher K-Theories (Lecture Notes in Mathematics). Vol. 341. Proceedings of the Conference Held at the Seattle Research Center of Battelle Memorial Institute. Springer, 1973, pp. 85–147. ISBN: 9783540064343.

Dear Professor Miller,

Here is my response:

In the construction of BC in section 1, its required that C be a small category. This is only up to equivalence, right? I should have in my mind that we want to apply this to the category of vector bundles on a space, right? But vector bundles over a point gives you the category of vector spaces, which isn’t small… But its equivalent to a small category where you just record the dimension of the vector space (which is obviously gives the isomorphism class). Is this what he means when he says “the proceeding definition extends to categories having a set of isomorphism classes of objects” on page 95/19/103 (depending on which page numbering system you’re looking at)?
I already have in my mind a notion of BG for G a group… But I can also define a group to be a category with one object and the morphisms indexed by elements of G with composition given by the group law. Do I get an equivalent classifying space under this construction?
We have a notion of K_i already, which is the K-homology (we can define using the theory of spectra in Lyubo’s paper). If we take the category of vector bundles on a topological space, do these “topological” and “algebraic” notions coincide at all?
These higher K-groups don’t seem to be easily computable (at least from what I can see in the first section of this paper). Have people come up with good techniques for computing higher algebraic K-theory? How useful are these K-theory groups?

(To be fair, I know that there is a seminar going on at Harvard at the moment on higher dimensional class field theory that uses algebraic K-theory, so I guess number theorists have found this useful!)

Thanks!
Isabel

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5 Nov

Sure - the category of finitely generated projectives isn’t small. But any two choices of small equivalent subcategories will give you canonically homotopy equivalent Q constructions.
Yes, of course. You could see it in the specific description of the nerve that [Student Name] gave. However, there are some differences. BC might not be connected! And any space is weakly equivalent to the classifying space of some category. (The category of simplices in the singular simplicial set, for example.)
The algebraic K-theory of the discrete ring of complex numbers is not quite right. For one thing, topological K-theory is periodic, while algebraic K-theory is the homotopy of a space…
This can be modified; the space BQ is actually an infinite loop space, so K_*(M) is the homotopy of a spectrum, but it still is zero for negative dimensions. And even in positive dimensions it’s not quite right. There is a way to put the topology of C back into play, and then you get BQ = BU.
Not easily computed is putting it mildly. Quillen (in a paper we discussed) computed it for finite fields. Borel computed it rationally for number rings. Quillen’s theorems in this paper describe relationships between K-theories of different rings. But some of the deepest work in topology in the past thirty years has been concerned with this question: The Quillen-Lichtenbaum conjectures, Voevodsky’s work on the Milnor conjecture, the entire development of the cyclotomic trace by Madsen, Hesselholt, and many others … all this is addressed at these computations. Clark Barwick represents the contemporary forefront of that effort.

Algebraic K-theory has a lot to do with algebraic cycles. Spencer Bloch is the relevant name. To be fair, I think it’s fair to say that the payoff in arithmetic has not been as great as had been hoped at first. But maybe it’s just a matter of time. I had not heard of this seminar on higher dimensional class field theory.

Haynes

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7 Nov

Response to: Nishida, G. “The Nilpotency of Elements of the Stable Homotopy Groups of Spheres.” Journal of the Mathematical Society of Japan 25, no. 4 (1973): 707–32.

Dear Professor Miller,

I’m sorry this is going to be a terrible response, since I spent most of my time over the past few days preparing my talk. But anyhow, some thoughts:

I think this paper is pretty cool, I like seeing that this “D_nX” construction we saw on the first day to construct the Steenrod operations comes up to prove this theorem about nilpotency of order p elements in \pi_t^s(S^0). I also like papers with concrete computations!

The proof of proposition 1.6 using the fact that D_n^(r)(S^k) is a Thom complex of a vector bundle which is a of finite order in the real K-theory of BS_n^(r)… I am a little confused about how the conclusion follows?

Also, a key theorem for him is the homology of the “infinite loop space” Q(X) =lim \Omega^\infty S^\infty X given in Theorem 2.3 due to Dyer-Lashof. The result clearly reminds one of the cohomology of Eigenberg-Maclane spaces that we already saw. What was the original motivation for proving such a result on the mod p homology of Q(X) in terms of the homology of X? (Obviously its very useful here.)

Thanks!
Isabel

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7 Nov

Isabel

Do you get the identification of D_n S^k with a Thom space? Remember the exercise I didn’t carry out? (It’s sketched, but poorly, by Hatcher.) You want to know that Sq^0 is nonzero in some example; why not S^1. So look at D_2(S^1) . The claim is that it’s the Thom space of the bundle over RP^\infty associated to the canonical representation of C_2. This representation is 1 + sign, so the Thom space is the suspension of RP^\infty. Now you have to think about what the diagonal map looks like in this presentation. It’s going to be the map induced by the inclusion of sign into 1+sign. This induces the map \Sigma RP^\infty_+ –> \Sigma RP^\infty sending the + to the basepoint. This is nonzero enough!

I guess you have to pick the skelata carefully: you want them to have torsion homology. Then the AHSS tells you that the K-theory will be torsion as well.

I guess the motivation for computing H_*(QX) is that it’s there! But the result is amazing: it’s a functor of H_*(X) (field coefficients). It’s worth thinking about what the answer is with Q coefficients; it’s simpler than the finite characteristic case Nishida needs. The key to the calculation is the approximation theorem; this relates QX to the n-adic constructions. So then the surprise is that H_*(D_nX) is a functor of H_*(X). This fact is almost a consequence of the construction of the universal power operation.

Haynes

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14 Nov

Response to: Quillen, D. G. Homotopical Algebra (Lecture Notes in Mathematics). Vol. 43. Springer, 1967. ISBN: 9783540039143.

Dear Professor Miller,

Here is my response to Quillen’s paper:

I think I have some trouble with these papers that introduce a new construction or concept with seeing how this is useful to solve problems/fits into a larger goal of understanding something. So I asked [Student Name] what the “main point” of this paper was and she said she thought it was the construction of the homotopy category (in terms of inverting weak equivalences) and Theorem 1 (or 1’) which gives a means of more concretely understanding and working in the homotopy category (something I see paying off for Quillen in the latter sections of the paper). She said another key result was the adjunction Theorem 2 at the end of Section 2 giving loops and suspension functors in the homotopy category (and more generally Theorem 3 on left and right derived functors of adjoint functors between model categories).

Would you agree with this? Maybe you could give me some references for how this formalism has paid off more recently (you don’t need to convince me that derived categories are useful, but this seems more general, and I this is my first encounter with model categories). Some things that I found enlightening were examples and remarks at the end of section 4 following the proof of Theorem 3 mentioned above. I should also probably look at the second half of the paper that [Student Name] isn’t planning on covering, because it might provide more motivation by way of examples.

Thanks,
Isabel

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16 Nov

Isabel

Just wait! [Student Name] will describe a wonderful example of the practical application of this theory.

But in general, you may want to be sure that some construction has homotopy theoretic content, or want to know how to modify it so that it does. This is how Tor is derived from tensor, or Ext from Hom. But what if you’re working in a non-additive situation, where chain complexes don’t make much sense (and where the proof of the fundamental theorem of homological algebra fails)? A good example is provided by commutative rings, but almost any category of universal algebras well work the same way. In the second part of HA, Quillen sets up a model category structure on simplicial obects in such a category. This lets him say what a “projective resolution” is: it’s a cofibrant replacement of a constant object. In another landmark paper, he explains what the universal meaning of “homology” is: you fix an object X and look at the category of abelian objects over it. There’s usually an abelianization functor Ab from objects over X to Abelian ones. Apply Ab to a cofibrant replacement for X as an object over itself; this gives you a simplicial object in a category of abelian objects, which is usually abelian. So now you can form the associated chain complex and its homology. That’s the “Quillen homology” of X.

In brief, Quillen homology is derived functors of abelianization.

Haynes

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19 Nov

Response to: Bousfield, A. K. “The Localization of Spaces with Respect to Homology.” Topology 14, no. 2 (1975): 133–50.

Hi Professor Miller,

Here is my response to the Bousfield paper:

Yes! This was a very nice application of Quillen’s model category framework! Especially since the main theorem is so easy to prove once you have that the h_*-equivalences can be used as weak equivalences instead of weak homotopy equivalences in the model structure on the category of simplicial sets. It definitely makes it seem like the right framework.
But I want to make sure I understand the proof that localizations exist with respect to any homology theory h_*. So he wants to prove that there is a localization functor E on the pointed homotopy category of CW complexes, along with a natural transformation \eta : 1 \to E satisfying a universal property. But to prove this, he uses the alternate model structure defined above on simplicial sets. This reminds me of one of the other theorems Quillen proved in the paper we read that there is an equivalence of model categories between the pointed homotopy category of simplicial sets and the pointed homotopy category of topological spaces via the adjoint functors of geometric realization and singular set. Does he not have to deal with any of the details of this because he can define this model structure on the pointed homotopy category of CW complexes by the structure on simplicial sets and geometric realization?

Is this what he means when he says “This abuse [of notation in letting Ho denote the category of Kan complexes or CW complexes] is harmless because the geometric realization provides an equivalence between the Kan and CW pointed homotopy categories.”? He cites a paper of May on “Simplicial Objects in Algebraic Topology” here.
Do people care about homology theories that aren’t ordinary singular homology with coefficients? (I would assume so… such as K-theory?) And homology theories that aren’t connective, which he says means that the acyclic spaces are the same as ordinary homology with coefficients? I couldn’t find much about this by some short googling…
Why are the “algebraic” conditions given in sections 5-9 useful? Is it just for intuition or completeness, not that they are actually computable?
Also googling around about this subject I found: http://mathoverflow.net/questions/173546/why-localize-spaces-with-respect-to-homology, which seems to say that localization can not only simplify a problem but make more phenomena appear… I also like the observation that the only reason Adams lacked a proof of localization with respect to any homology-theory was set-theoretic concerns.

Thanks!
Isabel

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19 Nov

Isabel

Glad you liked it!
Well, I think that all the model category work in this paper takes place in simplicial sets. He uses spaces only to say what he means by ‘generalized homology theory.’ I think he refers to may rather than Quillen at this point because he doesn’t even care that the homotopy *theories* of simplicial sets and spaces coincide; only that the homotopy categories do.
You bet! Yes, K-theory is a good example; we defined a cohomology theory, represented by a spectrum, and in turn the spectrum determines a homology theory, by taking the homotopy of a space smashed with the spectrum. In fact there’s a whole swarm of analogues of K-theory, also nonconnective, including so-called elliptic homology theory.

And there are many interesting connective theories. Bordism is one, determined by a Thom spectrum, and represented by bordism classes of singular manifolds. I think [Student Name] will talk about complex bordism, for example. Any spectrum admits a “connective cover,” in which the negative dimensional homotopy groups have been killed. So there’s “connective K-theory,” for example.
Well, localization can’t exactly create something from nothing. But things become so much clearer and easier to talk about that new things appear. Stable homotopy theory itself is a kind of localization; you are inverting the suspension functor. And there are are a lot of ’new’ things there!

Yes, Adams proposed this topological enrichment of Serre’s mod C theory, but neglected the set theory. I think he was also mainly interested in the stable picture. Bousfield has another paper called “Localizing spectra with respect to homology.”

Haynes

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20 Nov

Response to: Griffiths, P. A., and J. W. Morgan. Rational Homotopy Theory and Differential Forms. Birkh ̈auser, 1981. ISBN: 9783764330415.

Hi Professor Miller,

Here is my response to Griffiths and Morgan ([Student Name] told us to skim chapters 8–12):

I really enjoyed this paper!
I am glad that Prop 8.7 and Cor 8.8 hold! I was starting to type out a question to you about how C^\infty manifolds can be triangulated, so if I viewed this manifold as a simplicial complex by this triangulation, would I get the same homology theory taking piece-wise linear forms and C^\infty forms… but then I got to this proposition and I see that you do, in fact (up to tensoring with R). But you don’t get the same DGA of the cochain complex.
I really like section IX about commuting cochains and that the obstruction for Z-coefficients is really in the existence of cohomology operations (well, the Steenrod ops can be defined using failure of commutativity on the cochain level – this was a nice tie-up with things we’ve seen before)! But I am a little confused about why rational cohomology for a simplicial complex doesn’t give commutative cochains…? I don’t see why (v) isn’t satisfied —this is saying that the map on cochains isn’t surjective, right? But why can’t you extend your function from Y to X by 0 on the complement? I’m probably missing something really silly here…
I don’t really appreciate this sentence “So now we have come full circle. The problem of commutative cochains is equivalent to finding not only the cohomology, but also the Q-homotopy type of a space from a chain complex.” – probably from not reading the first 7 chapters of the book. But the idea is that we want to extract from Q-cochain data the maximum information, and it turns out that you can tell the Q-homotopy type if you have commuting cochains (but once you pass to homology you loose this). Is this the most important take-away of the paper – coupled with the piece-wise linear forms which realize commuting rational cochains? Because sections IX, X, XI, and XII seem to be devoted to understanding this.
Building on (3), is it hard to compute commuting cochains? (Or we could just say piecewise linear differential forms for concreteness.) Rational homotopy doesn’t include the information of torsion in the fundamental groups, so perhaps it’s more computable anyways?

Thanks!
Isabel

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21 Nov

Isabel

There are many DGA models (but up to noncanonical iso the minimal model is unique).
The “complement” isn’t a simplicial complex. Or: if you think of a 0-cochain as a continuous function to Q, you won’t be able to extend it continuously.
In other developments (Quillen, Bousfield-Guggenheim), you construct a model category structure on commutative rational coalgebras and show that it’s Quillen equivalent to the HQ-local model category on simply connected spaces. So DGAs give you a complete picture of rational homotopy theory.
Let’s try to find the minimal model of S^{2n}. [Student Name] didn’t quite say this explicitly, but part of his dictionary is that \Lambda(x_k) is the minimal model for K(Z,k). (You can check this by inductively computing the rational cohomology of K(Z,k).) Map S^{2n} –> K(Z,2n). It’s not a rational equivalence; there’s a 4-dimensional class in the EM space that pulls back to 0. So kill it by a map to K(Z,4n). The homotopy fiber sits in a fiber sequence K(Z,4n-1) –> E –> K(Z,4n), and now S^{2n} –> E is a rational equivalence. The corresponding Hirsch extension is \Lambda(x,y) with |x| = 2n, |y| = 4n-1 , and dy=x^2. That’s it!

The most amazing thing, from my perspective, is this: the rational homotopy of X is the vector space dual to the indecomposables in minimal model for X. Check: the rational homotopy of S^{2n} is nontrivial only in dimensions 2n and 4n-1.

There are general classes of spaces which are known to be “formal,” that is, the cohomology ring itself serves as a DGA model. Polynomial cohomology is one way to guarantee this. Also the famous theorem of Deligne, Griffiths, Morgan, Sullivan, says that any compact Kaehler manifold is formal.

Haynes

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24 Nov

Response to: Segal, G. “Categories and Cohomology Theories.” Topology 13, no. 3 (1974): 293–312.

Hi Professor Miller:

Here is my response to the Segal paper:

So the (rough) basic idea of this paper is that to a category we associate an infinite loop space (well, a spectrum which is often an infinite loop space). And this is a generalization of a topological abelian group because composition only commutes up to homotopy. The spectrum we get then defines a generalized cohomology theory, which again generalizes what we would get from a topological abelian group in which this is just the direct sum of ordinary cohomology with coefficients. Is that right? The obstruction to the cohomology theory being direct sum of ordinary cohomology is the fact that composition doesn’t commute on the nose so an infinite loop space doesn’t have the homotopy type of a topological abelian group? (This might be a really silly question, but is there some way to measure difference in cohomology theories that illustrates this obstruction to your theory being ordinary cohomology?)

Something else:

I was trying to think if his definition of classifying space of a Gamma space is natural in that it generalizes classifying spaces of topological abelian groups – I mean, it must, right? So I tried to work this out with my hands, if you took the Gamma-space corresponding to a topological abelian group (well, just finite abelian group to make it easier). I got a little stuck —but I’m going to try to work this out after seeing [Student Name’s] talk today. One potentially tricky thing is that he uses a different definition of realization..? Is it easier to work this out in the language of Gamma spaces or Gamma categories?

(By the way, I figured out an answer to the kernels and cokernels of locally free coherent sheaves… I’ll tell you today in our meeting.)

Thanks!
Isabel

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24 Nov

Isabel

Well, let’s see. A category has a classifying space. In fact any space is weakly equivalent to the classifying space of the category singular simplices in it. So you need extra structure to get an infinite loop space. One such structure is a symmetric monoidal structure. This still doesn’t quite give an infinite loop space, generally; you only get one after “group completing,” which is, in this topological context, formation of the loop space on the next space in the spectrum.

Mere homotopy commutativity isn’t enough, right? You need something to produce extensions of the n-fold product through the extended n-th symmetric power, and they have to be compatible. Or, in Segal’s framework, you need an entire \Gamma-space, not just the first couple of terms of one.

What you say about topological abelian groups and ordinary cohomology is right! The best answer to the question of how to distinguish spectra that are not products of EM spectra from those that are is that the K-invariants are nontrivial in the former case and trivial in the second. But you can also just inspect: Z\times BU for example is the 0-space of the omega spectrum for K-theory, and it’s visibly not a product of EM spaces.

Remember, the “classifying space” construction takes a \Gamma space to another \Gamma space. As [Student Name] pointed out, (BA)(1) is the geometric realization of A regarded as a simplicial set via the inclusion \Delta^op –> Fin_*. The \Gamma space A associated to an abelian group G has A(n)=G^n, and you can check that the structure of simplicial space is exactly the bar-construction simplicial space whose realization is BG.

I agree with your resolution of my confusion, but I still feel confused.

Haynes

1 Oct

Response to: Milnor, J. W. “On Manifolds Homeomorphic to the 7–sphere.” Annals of Mathematics 64, no. 2 (1956): 399–405.

Dear Professor Miller,

Here is my response to the Milnor paper:

A very key point to the argument is that any closed and oriented 7 manifold can be realized as the boundary of an eight manifold—e.g. the degree 7 piece of the cobordism ring is trivial (not just 2-torsions!)… After reading this I looked back at pg 203 of Milnor-Stasheff and sure enough he cites this result (as well as that the degrees 1, 2, 3, and 6 pieces vanish as well). But by taking appropriate products you can show that above 7 all cobordism groups are nonzero. So this method of generating an invariant of 7-manifolds is quite unique to this case. Presumably people have found other examples where homeomorphic manifolds of dimension higher than 7 aren’t diffeomorphic…. Do you know what extra ingredients had to go into this? Also, before this paper, did people think that homeomorphic smooth manifolds with a differentiable structure were diffeomorphic?
I found the Morse theory argument used to show the 7-manifolds constructed are ``exotic 7-spheres" very interesting if a little out of my comfort zone of things I understand (I know very little about this). Is this a standard way of proving the types of results that Milnor is after? (e.g. that if you have such a map with such critical points then you can cook up a homeomorphism onto some topological space). In his practice talk [Student Name] mentioned you might use homotopy theory to prove that these topological spaces constructed are exotic 7-spheres. A nice thing that comes out of this Morse theory approach is that the obstruction to being diffeomorphic to S^7 (with its standard differentiable structure) is a single point (I’m not sure if this is obvious otherwise?)

Thanks!
Isabel

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1 Oct

Isabel,

It helped a lot when Kervaire and Milnor showed that any smooth manifold homotopy equivalent to a sphere embeds in a Euclidean space with *trivial* normal bundle. That guarantees that it will bound, since all its SW classes are zero. It also gives an invariant in the group of framed bordism classes modulo reframings, the “cokernel of J.” K&M go on to almost completely analyze this; In dim 4k it’s an iso. In dim 4k-1 (eg 7) it is surjective with kernel determined essentially by the signature theorem (so cyclic of order given in terms of Bernoulli numbers). In the other dimensions they nail things down up to a factor of two, which is the famous Kervaire invariant, subject of this semester’s Juvitop.

People worked very hard to prove that there were no exotic smooth structures in dimension 2. They understood the possibilities, and spoke of the “smooth Poincare conjecture.” But I don’t think there were fixed beliefs. Certainly Milnor’s paper was celebrated!

[Student Name] wanted to apply the “h-cobordism theorem,” which I think came after this paper. It gives a systematic way to try to prove that two homotopy-equivalent manifolds are diffeomorphic, and comes with an obstruction (the “surgery invariant”) if it fails. But this is out of my comfort zone too!

Haynes

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3 Oct

Response to: Atiyah, M. F. K–theory (Advanced Book Classics). 2nd ed. Notes by D. W. Anderson. Addison Wesley Longman Publishing Company, 1989. ISBN: 9780201093940.

Dear Professor Miller,

Here is my response:

There seem to be several ways to define K-theory of X (say complex K-theory)…. you could do it as maps from X to BU(\infty). Well, this gives you K^0(X) reduced. To get higher homotopy groups you’d have to map into a space whose loops is the classifying space for n-1st K-theory. Then you apply Bott periodicity to see that the homotopy classes of iterated loops on BU(\infty) are 2-periodic, and so K-theory is periodic, and we really care about K^0 and K^1… Or you could take the route of defining higher K-groups to be the derived functor of K^0… Or you could do what the paper does and say that (Def 2.4.1 on pg 68) that reduced -nth K theory is the reduced K-theory of S^nX. Its clear that some of these definitions agree, like the third implies the second because Atiyah proves the LES in K-theory. And maybe the second implies the first by general theory. Its also pretty easy to see that reduced K^0 agrees with this Maps(X, BU), once you have the results that Atiyah covers in section 1 about any k-plane vector bundle as the subbundle of a trivial bundle, you naturally get the map to the Grassmanian of k-planes. I guess I’m wondering about the relation between these definitions and what is most useful to think about in practice?

A related question is what tools are there to compute K-theory… I know there is the spectral sequence relating the K-theory of a space to its cohomology and the K-theory of a point… which is easy. You could also use the fact that the chern classes give maps from K-theory to cohomology. Atiyah manages to use the LES in K-theory as well around pg 80. What other tools are there?

Thanks,
Isabel

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3 Oct

[Student Name] did a nice job of linking up the various definitions.

Actually a ‘reduced’ cohomology theory is best thought of as K(X,*). So it’s *pointed* homotopy classes of pointed maps X –> Z x BU. If X is connected (as [Student Name] kept assuming) then a pointed map has to land in the 0-component of Z x BU (which is where the basepoint is). But if X is not connected, then reduced K-theory is not quite [X,BU].

And it’s not quite true that \Omega^(BU) = BU: rather, it’s Z x BU.

Hey, how can I think of higher K-groups as derived functors of K^0? Seriously, if you have a way of thinking about that please tell me.

Atiyah basically doesn’t mention cohomology; this is the axe that he is grinding. But a major tool in understanding K-theory is the AHSS, which you mention. Another is the Chern character,

ch: K –> HQ. It’s a multiplicative transformation, sending K^0(X) to the direct sum of the even rational cohomology groups of X (for X a finite complex) and K^1(X) to the sum of the odd groups.
It’s a rational isomorphism….

Haynes

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8 Oct

Response to: Adams, J. F., and M. F. Atiyah. “K–theory and the Hopf Invariant.” The Quarterly Journal of Mathematics 17, no. 1 (1966): 31–38.

Dear Professor Miller,

Here is my response to Hopf Invariant 1:

The main thing one has to note from this paper is how simple and beautiful the argument is! (But I think that’s most apparent, from my perspective, since Adams and Atiyah didn’t have to do any difficult computations of K theory – the reduced K-theory is easily seen to be free on the generators coming from the cells in dimensions n and 2n in cohomology… Then if you compare the arguments in the vector fields on spheres paper and this one (modulo the computation of K-theory) they are both very simply and boil down to very similar obstructions.) Something I was wondering was, when you have an isomorphism between the K-theory and cohomology rings, what do the Adams operations correspond to on the level of cohomology? Is this even a reasonable question to ask?

Could we plan on meeting this afternoon? You can let me know when you’re free after 4:30 (alternatively I can meet right after class until 11:30 am or before class).

Thank you,
Isabel

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8 Oct

Isabel

You are only guaranteed a ring isomorphism modulo filtration, or rationally. The rational bit comes from the Chern character, a ring homomorphism ch : K(X) –> H^{even}(X;Q) that extends to an isomorphism after tensoring with Q. Under ch, \psi^k corresponds to multiplying by k^n on H^{2n}: so it picks out the grading as eigenspaces.

Somehow the proof works for K-theory because you are entitled to work modulo a higher power of 2. K-theory has some surprising torsion properties; for example, the K-theory of RP^\infty is torsion-free.

See you this afternoon.

Haynes

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20 Oct

Response to: Quillen, D. G. “The Spectrum of an Equivariant Cohomology Ring: I.” Annals of Mathematics 94, no. 3 (1971): 549–72.

Dear Professor Miller,

Here is my response:

The results about equivariant cohomology of G-spaces seemed very natural, when you think that for X with a free G-action, the G-equivariant cohomology should just be H^*(X/G). But then the problem is that if the action isn’t free, this isn’t a homotopy invariant (e.g. S^\infty and a point with a Z/2Z action, which is free on S^\infty, but the quotient RP^\infty is not contractible). So if you want to define equivariant cohomology of X to be the cohomology of Y/G for some Y in the homotopy class of your space with a free G action, then EG \times X would give such a space Y. And this is exactly the definition Quillen gives (except to call it PG).

I found the results about the cohomology of BG (e.g. G-equivariant cohomology of a point) very interesting. In particular theorem 7.1, which says (approximately) that to calculate H^*(BG, Z/pZ) up to nilpotents, it is enough to know H^*(BA, Z/pZ) for A an elementary p-group. In the case that G is a finitely generated abelian group, BG is a K(G,1). And Serre’s paper (which [Student Name] also presented!) gives techniques to compute the cohomology of this Eilenberg-MacLane space. So can you prove these results easily using this machinery in this special case? Or even in the special case when p=2 and you can just apply Serre’s results? In any case, this Theorem seems interesting, as well as the corollary that the dimension of H^*(BG, Z/pZ) is the maximum rank of an elementary p-group. What’s the intuition behind why this theorem (7.1 or more generally for G-spaces) should be true?

Thanks,
Isabel

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20 Oct

Isabel

BG is always a K(G,1) … you can define H^1(X;G) for non nec abelian G (as isomorphism classes of principal G-bundles over X).

But we’re speaking here about K(Z/p,1) (or a power of this space). That’s easy; its skelata are called “lens spaces.” One way to get at it is to think about the fibration derived from the group extension Z/p –> S^1 –> S^1: S^1 –> BZ/p –> CP^\infty. With integral coefficients, p times the generator of H^2(CP^\infty) pulls back to zero, so you must have d_2(generator of S^1) = px. From this you can conclude that H^*(BZ/p;F_p)=E[e]\tensor F_p[x] with |e| = 1, |x| = 2, and the two connected by the Bockstein. (When p=2 there’s an extension, e^2=x.)

So Spec H^*(BE) is an affine space, if E is elementary abelian. Quillen shows that Spec H^*(BG) is a colimit of these affine spaces; it’s a bunch of affine spaces, corresponding to conjugacy classes of maximal elementary abelian subgroups, glued together according to certain conjugacies. In particular the Krull dimension of H^*(BG) is the largest of the ranks of the elementary abelian subgroups.

Maybe the intuition comes from the corresponding result where you replace F_p by Q. Then it goes back to Elie Cartan, maybe, that H^*(BG;Q) = H^*(BT;Q)^W . Here G is a connected compact Lie group T a maximal torus, and W the Weil group (ie the group of isomorphisms of T induced by conjugation by elements in G). The mod p picture is more complicated in general because there may be difference conjugacy classes of maximal elementary abelian p-subgroups. (They can even have different ranks.)

Haynes

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24 Oct

Response to: Wall, C. T. C. “Finiteness Conditions for CW–complexes.” Annals of Mathematics 81, no. 1 (1965): 56-69.

Hi Prof. Miller,

Here is a response to Wall’s paper:

If I understand correctly, a key insight of this paper is in the algebraic conditions placed on a complex to be equivalent to one of finite type (or finite dimension). That is, these conditions Fn for finite type, and Dn for finite dimension. At least in the case of finite type, the proofs seem to be rather straightforward and builds on Milnor’s construction in the simply-connected case.

Section 3 seems more interesting, in particular Theorem F (and the Lemmas 3.1 and 3.2 leading up to Theorem F), that gives that the class of \pi_n(\phi) in the projective class group for \phi an (n-1)map from a finite complex K^(n-1) to X depends only on X if X satisfies the algebraic conditions Fn and Dn. This really uses the fundamental group in a very nontrivial way, since if X were simply connected, then \Lambda = Z and as Z is a PID, projective Z-modules are free so this projective class group is trivial, and all classes are trivial [In fact as observed latter the same is true if \pi is free abelian]. Is it clear immediately that \pi is the obstruction to this problem being obvious? (I guess this comes up in section 1 as well, with the algebraic conditions Fn, and his key insight is extending Milnor’s work to the non-simply connected case).

Another question is whether the other questions he poses in the introduction have been answered? Is the best way to think about these finite conditions still in terms of the algebraic constraints put on classes in a projective class group?

Thanks!
Isabel

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24 Oct

Milnor’s theorem (in the appendix) is one of my favorite results! I agree, this paper is really working out what you can say in analogy with that result in the non simply connected case.

My impression of the utility of this result today is this. One receives a homotopy type, which you know in advance is finitely dominated. Often your ultimate goal will be to show that it contains a compact manifold. A first step is to show that it contains a finite complex. So those obscure conditions are not used to verify finite domination, but rather to use finite domination to construct the invariant in the projective class group.

One of the surprises of the construction of this invariant is that it depends only on the top dimension. I suspect that there is a different way to produce the invariant which is more intrinsic - not depending on the construction of a map \phi - but in exchange uses more dimensions. For example you have the chain complex of the universal cover …

Haynes

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29 Oct

Response to: Adams, J. F. Stable Homotopy and Generalised Homology (Part III) (Chicago Lectures in Mathematics). University of Chicago Press, 1995. ISBN: 9780226005249. [Preview with Google Books] [reprint of the 1974 original]

Dear Professor Miller,

Here is my response:

A disclaimer is that I know very little stable homotopy theory already, so I am mostly searching for some motivation. I see how the category of spectra is a natural definition if you want to study stable phenomena, and how the definition of maps in this category are the natural choice again if you care primarily about stable phenomena. I’m not sure exactly why we care about this (other than maybe its more computable?), and Adams hints that you can also use this to study unstable problems? And the stable category of spectra is sufficiently “nice”… Maybe you could also give me some context for this, since I don’t have much of a background in which to fit this. A couple of things that came up in the paper:

It’s a very nice argument that the spectrum associated to any cohomology theory (where you have excision) is an \Omega-spectrum. I think that Adams says that any (CW?) spectrum is equivalent to an \Omega-spectrum, so you can think of any spectrum as being iterated loop spaces. Is there an easy way to see this?

Then, we can defined a (co)homology theory using an \Omega-spectrum E with, like E^r(X) = [X, E]_r degree r maps… So maybe I should think about spectra as really ways to getting cohomology theories.
How do you explicitly define a degree 1 “map” (not function) of S \to S (the sphere spectra) that extends the Hopf map S^3 \to S^2? This isn’t going to be a fibration with sphere fibres, and the homotopy groups \pi_{n+1}(S^n) don’t stabilize until after the Hopf fibration at n=2. But Adams seems to hint at this ideas as a goal when motivating the discussion of maps not functions.

Thanks! I think some motivation will probably be provided in the talk tomorrow and in the future.

Best,
Isabel

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30 Oct

Isabel

Stable homotopy theory is a certain kind of localization of ordinary homotopy theory: you formally invert the suspension functor. It takes a certain amount of care to make this work right. So in a certain sense everything you do in the stable category is some sort of complicated limit of unstable constructions. But keeping track of the range of dimensions in which certain constructions work is a big hassle, and it’s taken care of automatically by the stable category. You saw an example in the construction of the Atiyah-Hirzebruch spectral sequence.

But then it turns out that this particular localization has extraordinarily beautiful properties, and makes a compelling object of study on its own merit.

It also forms a universal example. Any kind of derived category is enriched over spectra, for example.

Does this help? Here’s another motiviation: You are right, of course, spectra “are” cohomology theories. What the category of spectra (and maps) does for you is that it lets you form things like cofibers. So any natural transformation of cohomology theories embeds into a long exact sequence in which the third term is also a cohomology theory …. Coefficient sequences are one example of this.

Omega spectra … well, I can form \Omega^n E_{n+r} ; and by adjointness it maps into \Omega^{n+1} E_{n+r+1} ; so I can take the direct limit space … this wants to be the r-th space in the \Omega spectrum for E. There are probably some point-set topology problems there, which were probably dealt with by Peter May somewhere.

Hopf map … Can’t I just use \Sigma^{r-2} \eta : S^{r+1} –> S^r ?

Haynes

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31 Oct

Response to: May, J. P. The Geometry of Iterated Loop Spaces (Lectures Notes in Mathematics). Vol. 271. Springer-Verlag, 1972.

Dear Prof. Miller,

Here is my response:

So the motivation for replacing operads by monads is that the notion of monads is simpler to work with, and we have the Bar construction to get a simplicial space out of a monad…? Could this have all be phrased working with operads? (I guess, an operad determines a monad by construction given in Construction 2.4 of May’s paper,. But a monad is more general, right? So you can’t necessarily go in the other direction?) Lemma 2.10 saying that monads encode the information of an adjunction is a nice way to think about this. And its useful because it gives that \Omega^nS^n is a monad.
In the discussion on the bottom of the first page of chapter 5 (pg 39), when he says that the data of L^n (the sequence of n iterated loop spaces of Y) “serves to record the fact that Omega^nY does not determine Y and we must remember Y…” he means that we’ve lost information about the first n-1 homotopy groups of Y by taking n-fold loops, so we can’t uniquely go back to Y from \Omega^nY, right? But the whole idea is we can product some Y via delooping to show that \Omega^n Y is an n-fold loop space.
I don’t really see the difference between his definition of L^\infty and bounded \Omega-spectra? (First I’m not really sure what bounded means, and a quick check on the internet didn’t help). Is this something having to do with that he writes Y_i = \Omega Y_{i+1} in J for spaces in the sequence L^\infty, where as for an \Omega-spectrum we have only homotopy equivalences f_i : Y_i \to \OmegaY_{i+1}?

Thanks!
Isabel

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31 Oct

Isabel

Yes. The monads (or in an older language the triples) you get from an operad are special. For one thing, you need a symmetric monoidal structure just to make sense of the definition of an operad. Also the construction makes them behave smoothly - commute with filtered colimits, for example. I’ve wondered whether there’s an obstruction theory somewhere here actually.

By the way [Student Name] didn’t point this out but the use of operads in other symmetric monoidal categories is extremely important; I think Peter’s failure to set them up in that generality in this paper somehow hid them from the algebraists for many years.

Going to the monad determined by an adjunction does forget information. You get one of the categories by design. For the other, one choice would be the category of algebras over the monad. An adjoint pair is “tripleable” if that category of algebras is actually equivalent to what you had to start with. Sometimes yes, sometimes no. There is more than one adjoint pair giving a monad. The category of algebras is one extreme. The other extreme is the “Kleisli construction,” which you can google.
I think it’s best to say that an “n-fold loop space” IS an expression of the space as the n-fold loops on a specific other space.
Sorry, I don’t know the reference here. In other work (maybe here too) Peter May proposes to study (X_i, e) where e: X_i –> \Omega X_{i+1} is a Homemrphism. He calls these “spectra,” and the things with just maps X_i –> \Omega X_{i+1} he calls prespectra.

If you point out the specific reference I can try to respond.

Happy Halloween
Haynes

4 Sep

Response to: Steenrod Operations (PDF - 1.7MB)

Dear Professor Miller,

Here is my response:

Supposed X is a smooth manifold. Then any cohomology class in H^q(X) corresponds under Poincare duality to a linear combination of codim q submanifolds. For u_Y a class in H^q(X) corresponding to a cidim q sub manifold Y, you can compute u_Y \cup u_Y as the cohomology class of the self-intersection of this submanifold—e.g. as the cohomology class of the zero locus of a section of the N_(Y/X) (or as the index q Stiefel-Whitney class of the normal bundle).

I’ve heard that Sq^i(u_Y) corresponds to the index i Stiefel-Whitney class of Y. This gives me more intuition for these cohomology operations.

Is there a way to understand this in terms of the construction you gave?

Thanks!
Isabel

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7 Sep

Dear Isabel,

Haynes

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9 Sep

Response to: Serre, J. -P. “Cohomologie Modulo 2 des Complexes d’Eilenberg-Maclane.” Commentari Mathematici Helvetici 27, no. 1 (1953): 198–232.

Dear Prof. Miller,

Here is my response:

One of the things that stood out to me in this paper was how clean the computations of the cohomology ring mod 2 of K(Z/2, n) was. In particular, the fact that the Serre spectral sequence could determine this without computing any of the differentials (although this may be built into Borel’s theorem, so maybe this isn’t quite true).

A question this brings to mind is: does this technique generalize to mod p coefficients? This would be relevant to the P^i cohomology operations. From what I can tell, the only new input would need to be an analog of Borel’s Theorem (which would come with some notion of a simple system of generators). Presumably this could classify operations of type (Z/p, Z/p, n, q) by Serre’s notation. (And probably has been done…? I would be curious if it also used this spectral sequence technique.)

Isabel

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11 Sep

Well, now … it depends on what you mean by “compute.” As [Student Name] pointed out, if you just started to compute H^*(K_2) from the spectral sequence, you’d have a tough time. You’d start by computing (if I may use that word) d_2 i_1 . Then somehow you would successively compute d_3 i_1^2, d_5 i_1^4, d_9 i_1^8, …. This is a pretty complicated spectral sequence, one which doesn’t collapse at any finite stage. Once you’ve done the first computation, it’s true that all the rest is built into Borel’s theorem.

By the way, if you reorganize the differentials to be done all at once, what you have is a contractible complex of … well, of comodules over H^*(K_1) as a coalgebra. You have a resolution of F_2 as a comodule. The comodule primitives of the comodules in that resolution are given by the bottom row in the spectral sequence. This shows that H^*(K_2) is the derived functors of H^*(K_1)-comodule primitives applied to the comodule F_2. Sanity check: the algebra dual to H^*(K_1) is exterior on classes in dimension 1, 2, 4, 8, …. Each gives a polynomial algebra in Ext, and the dimensions of the generators come out to be 2, 3, 5, 9, ….

Sure, the odd prime picture comes out just the same. The hardest thing is constructing the operations, which Hatcher describes very nicely. ‘Simple system’ means you can use powers up to the (p-1)st power of even generators.

Haynes

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11 Sep

Response to: Milnor, J. W. “The Geometric Realization of a Semi–simplicial Complex.” Annals of Mathematics 65, no. 2 (1957): 357–62.

Dear Prof. Miller,

Here is my response:

Something that confused me when I was first reading the paper was the definition of the face and degeneracy maps of K and delta_n, until I realized that on the level of the standard simplices, the face maps gave inclusion of an n-simplex as a face in an (n+1)-simplex and the degeneracy maps gave a projection from an n-simplex to some (n-1)-simplex (which is something of the opposite of what is happening with what I think of as the face and degeneracy maps of a simplicial set). Once I sorted this out, though, the notion of geometric realization he defines seems intuitive.

I guess I am wondering what is significant about this paper (or what is the key point to take away?) It seems that the notion of geometric realization was already defined, but I gather from the introduction that it doesn’t give the desired nice properties when dealing with products. (Perhaps what’s nice is the point that |K(pi, n)| is an abelian topological group? E.g. the space realizes the fact that something in the homotopy class of any K(pi, n) space will be a topological group.) I still feel the lack of motivation for this. Perhaps this will be filled in in [Student Name’s] talk tomorrow.

Isabel

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11 Sep

Dear Isabel,

This adjoint pair is a very general construction. If D is a small subcategory of C, then any X in C defines a functor D^{op} –> Sets by d |–> Hom_C(d,X) . If C has colimits then this functor has a left adjoint, formed |K| = \coprod_d K(d) / an equivalence relation defined by the morphisms in d.

Part of what I wanted this paper to do was simply to introduce the idea of simplicial objects. But I think the main theorem is that |Sin(X)| –> X is a weak equivalence: it’s a functorial resolution of X by a CW complex. If you define a map of simplicial sets to be a “weak equivalence” if it induces a weak (or equivalently homotopy) equivalence on realizations, then it follows that K –> Sin|X| is also a weak equivalence. This is saying that for homotopy theoretic purposes you can work in either category.

A main reason for using degeneracies is just what you say: they make products work correctly.

We’ll see whether [Student Name] describes the Eilenberg model for K(pi,n). Here it is:

K(pi,n)_* = Z^n(\Delta^*,pi), n-cocycles on the standard simplicial simplex with coefficients in pi

\Delta^n is the simplicial set \DDelta(-,n) , where \DDelta is the category with objects
[n] = {0,1, …, n}, n \geq 0, and order preserving maps.

So the realization of this is a topological abelian group model for K(pi,n). (pi abelian of course)

Haynes

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15 Sep

Response to: Thom, R. “Quelques propriétés globales des variétés différentiables.” Commentarii Mathematici Helvetici 28 (1954): 17–86.

Dear Professor Miller,

Here is my response:

First, could you give me a reference for the construction of the Thom class (and the Thom isomorphism…is this possible?) from the relative Serre spectral sequence for the fibration defined by the vector bundle? [Student Name] touched on this briefly in his practice talk and I was not very clear on what was happening. I also can’t see quite how to get Thm 8.1 and the preceding cohomology groups on pg 89 from the Serre spectral sequence.

From what I can see we take the relative Serre spectral sequence converging to the cohomology H^n(D(E), S(E)), where D(E) is the total space of the disk bundle of the n-plane bundle and S(E) is the sphere bundle (which is, I think, the same as taking H^n(E, E_0) where E_0 is the nonzero elements): then the E_2 page has 2 collections of points along the q=0 and q=n lines because the cohomology of the fibers is zero outside these degrees. It would seem, then, that the H^i(B) terms on the E_2 page with p<= n all survive to the infinity page (assuming they are nonzero), and it is only the higher terms that could be in the image of differentials. But this seems to contradict what is on pg 89. Also: how does a spectral sequence give you an identification of H^i(E, E_0) with H^(i-n)(B)? It seems like (at least a quotient of) H^(n)(B) would be identified with a graded piece of H^n(E, E_0)… I know that I must be doing something very silly… but I can’t seem to figure out what it is… (e.g. the i-n index looks something like working in homology and then using Poincare duality?)

Thanks!
Isabel

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15 Sep

Thanks, Isabel.

In this case the relative Serre spectral sequence has the form

E_2^{s,t} = H_s(B;H_t(F,F_0)) ==> H^{s+t})(E,E_0)

Well, H_t(F,F_0) is nonzero only for t=n, so there’s at most one nonzero row. There might be local coefficients. A trivialization of this coefficient system is precisely an orientation of the vector bundle. Every bundle is uniquely oriented with mod 2 coefficients, so we find the only nonzero row is

E_2^{s,n} = H_s(B)

The spectral sequence collapses; there is only one nonzero associated quotient group in each dimension; and so there is a canonical isomorphism

H_{s+n}(E,E_0) = H_s(B)

Under it, u corresponds to 1. Restricting to a fiber shows that this u restricts to the generator. The multiplicative structure of the spectral sequence guarantees that the isomorphism is given by cupping with u.

I wrote this before reading your second paragraph; does it answer your question?

The Serre spectral sequence leads to an easy proof of the splitting principle too: It’s enough to split off a single line bundle. The universal way to do that is to pull back to the projective bundle, so we want to see that the projection map from the projective bundle is monic. Look at the Serre spectral sequence; the fiber is RP^{n-1}; again no issue of orientability mod 2; by the multiplicative structure we just have to see that the generator a in E_2^{0,1} survives; and an explicit geometric construction shows that it does. What’s that construction?

Haynes

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17 Sep

Response to: Milnor, J. W., and J. D. Stasheff. Characteristic Classes (Annals of Mathematical Studies). Vol. 76. Princeton University Press, 1974. ISBN: 9780691081229. [Preview with Google Books]

Dear Professor Miller,

Here is my response:

The main theorem of Chapter 2 is really how I want to understand (co)homology of smooth manifolds – in terms of fundamental classes of submanifolds. (This of course related to my intuition about the Chow ring of an algebraic variety). If I understand this main theorem correctly, then this is well-founded if I am considering (lets do mod 2 coefficients) a submanifold of dimension < half the dimension of the manifold. So transferring this to cohomology, we have that that cohomology classes of degree > half the dimension of the manifold correspond (via poincare duality and the above) to submanifolds of codim = deg (=> dimension = n-deg which by assumption is < n/2).

Question: are there conditions upon the manifold under which this is true of all dimensions (up to taking linear combinations, such as in the integral coefficient case)?

(Is this obvious from the condition of the manifold being orientable in the Z case)?

e.g. Is the converse true under general condition?

My other question is, where does the obstruction to this being true in general come from in the problem? My first thought would be that you have a nontrivial normal bundle, but in the case that G reduces to a point, Theorem II.4 on pg 146 seems still to suggest this condition on the dimension… (silly question: how did the “and” in Thm II.3 turn into an “or” in Thm II.4?)

I am also a little confused about the appearance of the Stiefel-Whitney classes in the cohomology of Grassmannians but maybe this will be cleared up by the talk today.

Thank you in advance!
Isabel

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18 Sep

Great questions. Thom answers them in the way topologists answer such questions: he reduces them to calculations.

Take the unoriented case for example. So a in H_k(V^n;Z/2) is representable by a manifold iff its Poincare dual x is “representable.” This means that the map x : V –> K(Z/2,n-k) lifts to a map V –> MO(n-k). If n < 2(n-k) we are in the stable range; MO(n-k) is a product of Eilenberg Mac Lane spaces; the Thom class u : MO(n-k) –> K(Z/2,n-k) splits, so there is a lift and there is a submanifold. But if n > 2(n-k) (or maybe equal) then more complicated parts of the homotopy type of MO(n-k) get involved. One thing to check first is whether such a lift is compatible with Steenrod operations. It becomes a calculation. We could try to work out some examples. The oriented case is similar, and in fact the almost-complex case is similar too, maybe more relevant to complex manifolds.

The cohomology of real Grassmannians is made up of SW classes…. did this get clarified?

Haynes

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22 Sep

Dear Prof. Miller,

Here is my second response on this paper:

Since I already responded to the sections of the paper that will be covered on Monday, I will build on my response from last week.

I worked out clearly how he gets that the classifying space for O(k) is G_k, the Grassmannian of k-planes in R^m for m sufficiently large (let’s just say R^\infty). This comes from the fact that E_O(k) is the Steifel manifold V_k(R^\infty) of k-frames in R^\inft—clearly the O(k) action permutes these k orthogonal vectors transitively and the kernel of this action is hence G_k. So it suffices to show that V_k(R^\infty) is contractible. But this follows from induction on k… when k is 0 it is obvious. Then we use the fact that we have a fibration V_{k-1}(R^\infty) \to V_k(R^\infty) \to S^\infty, for which the last map is taking the last/first element of the k-frame. The space V_{k-1}(R^\infty) is contractible by assumption… but so is S^\infty, so using the LES in homotopy associated to this fibration we get the desired result.
I’ve cleared up my understanding of the cohomology of Grassmannians in terms of Steifel-Whitney classes. I see that these SW classes are associated to the obvious k-plane bundle on G_k, and I looked at an example of relating this to Schubert cells in the case of k=2, m=4. This cleared things up for me substantially.
I’d like to understand better what happens outside of the stable region for n > 2(n-k) and what the Steenrod operations might tell you (maybe we could talk about this). I’d like to get some intuition for what goes wrong when this realization question isn’t true.

Thanks!
Isabel

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22 Sep

Yes, terrific. This also shows why G_{k-1} is the sphere bundle of E_k –> G_k , more directly than what [Student Name] did today: the sphere bundle is V_k x_O(k) S^{k-1}, but S^{k-1} = O(k)/O(k-1), so V_k x_O(k) S^{k-1} = V_k/O(k-1) . The O(k-1) action is free (since the O(k) action was), and V_k is still contractible, so this is just another model for G_{k-1} .
The Schubert cell picture is beautiful. The actual ring structure of Gr_k(R^n) (or the complex version) is very complicated, though the Poincare series isn’t so bad. Actually you can generalize this computation in two ways: you can compute the cohomology of the flag manifold of decompositions of R^k into subspaces of dimensions k_1, …. k_n ; and you can do this on each fiber of a vector bundle and describe the cohomology of the flag bundle.
I tried to think through a little of this in my last response. Do you agree that any 2-dimensional cohomology class in an oriented compact manifold is Poincare dual to a submanifold? But we should be able to come up with counter examples in higher dimension, even without assuming orientability. There are more detailed questions, such as: what is the minimal genus of a surface representing a given class in a 4-manifold. This is very hard and geometric ….

Haynes

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26 Sep

Response to: Milnor, J. W., and J. D. Stasheff. “Signature Theorem.” In Characteristic Classes (Annals of Mathematical Studies). Vol. 76. Princeton University Press, 1974. ISBN: 9780691081229. [Preview with Google Books]

Dear Professor Miller,

Here is my response for the signature theorem reading:

The first thing I thought of while reading this was this “What is…” article someone who does differential geometry sent (attached) me when I told them I was writing my thesis on formal group laws [not that I still really understand the connection]. It gives a really nice introduction to multiplicative genera and how Hirzebruch’s theorem fits in. I like the motivation that this article as well as the reading from Milnor&Stasheff gives that such genera (well, the reading only says the signature) are depend only upon the cobordism class of the manifold, and that all homomorphisms from \Omega_* to Q are Q-linear combinations of Pontrjagin numbers (and hence can be given by the “multiplicative sequences” K that Hirzebruch defines), and so it makes sense that we should be able to find such a sequence for the signature.

I can see that this “multiplicative sequence” L that Hirzebruch defines for the signature is very related to the “logarithm” of the signature that is defines in that AMS article (well, clearly, since one is \sqrt(t)tanh^{-1}(\sqrt{t}) and the other is tanh^{-1}(t)), but I need to wrap my head around this a little more to see how exactly (I am certain that after class today I’ll have a clear enough picture to see this, its just unwrapping definitions.)

From the other chapters of reading, I would really like to understand the proof of Cor 18.9 that over Q the oriented cobordism ring is generated by the classes of the P^{2k}(C), but this was spread over many chapters and I didn’t spend long enough to completely trace through the argument in detail. I understand that the key steps are (1) showing that the 4nth degree of the cobordism ring is of dimension p(n), and (2) showing that the classes of P(C)^{2i_1} x… x P(C)^{2i_r} are independent as i_k range over all partitions of n. This last point comes from looking at characteristic classes, but I need to spend more time on this to understand it completely.

Thanks!
Isabel

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26 Sep

Dear Isabel,

I’d forgotten […] article. Earlier Graeme Segal wrote a Bourbaki Seminar note on elliptic cohomology with a similar summary of this story. I was urging [Student Name] to make the relationships a little clearer, but his focus was different. I could have referred to Hirzebruch’s book for this story, too.

As you know, there’s another perspective on (weakly almost) complex genera, which lets you think about ring homomorphisms into rings which are not Q-algebras: a homomorphism MU_* –> R is the same thing as a formal group law over R. It’s logarithm is the ’logarithm of the genus.’ The genus factors through MSO_* iff the corresponding formal group law enjoys a certain symmetry, which causes the odd coefficients in the logarithm to vanish. The hyperbolic tangent occurs by symmetrizing the exponential ….

If you are willing to use the fact that MU_* is the Lazard ring, then you know that its rationalization is generated by the coefficients of the logarithm, because over a Q-algebra any formal group is isomorphic to the additive formal group. So knowing “Miscenko’s theorem” (the formula Ochanine gives for the logarithm) tells you that the CP^n’s generate MU_* rationally. The MSO case is the “even” part of that statement.

But yes, this story is worth spending more time on!

Haynes

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