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There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on the moduli space of elliptic curves. I gather that the homotopy theorists regard it as a major achievement.

My question is as follows : what sorts of more classical topological problems is it useful for solving? It is clear why non-algebraic-topologists should care about the more classical extraordinary cohomology theories (e.g. K-theory and cobordism theory); they allow elegant solutions to problems that are obviously of classical interest. Does TMF also allow solutions to such problems, or is it only of technical interest within algebraic topology?

My background : my research interests are in geometry and topology, but I am not an algebraic topologist (though I have used quite a bit of algebraic topology in my research).

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I'm not sure about solving classical topological problems. The reason that I consider it to be interesting is because it's supposed to be to the second chromatic level (of chromatic homotopy theory) what K-theory is to the first chromatic level. The reasonn, in my mind, that this is interesting, is because the chromatic picture in stable homotopy theory happens to look a lot like the derived category of a Noetherian ring. That's sort of vague, but I think it speaks to some of the deepest structure of algebraic mathematics. –  Jon Beardsley Jan 1 at 4:51
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@JonBeardsley : I have to admit that I do not know what chromatic homotopy theory is, though certainly I have seen the buzz-word used before. I also have to admit that I am more impressed with theorems than with analogies, which is why I phrased the question the way I did. –  Doug P Jan 1 at 4:55
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You should swing by the homotopy theory chat room sometime. Those guys in there chat a lot (and know a lot) about TMF. It's not really my specialty. chat.stackexchange.com/rooms/9417/homotopy-theory –  Jon Beardsley Jan 1 at 5:05
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@JonBeardsley : I'm an old man; while I am willing to be "hip" enough to post on MO, I have no desire to engage in online chatting (I also don't text, tweet, or use Facebook). If they have interesting things to say, presumably they will show up here and say them eventually. –  Doug P Jan 1 at 5:10
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Brief side question: Why are homotopy theorists so much hipper than the rest of us? –  JamesE Jan 1 at 9:50

2 Answers 2

up vote 21 down vote accepted

One of the closer connections to geometric topology is likely from invariants of manifolds. The motivating reason for the development of topological modular forms was the Witten genus. The original version of the Witten genus associates power series invariants in $\mathbb{C}[[q]]$ to oriented manifolds, and it was argued that what it calculates on M is an $S^1$-equivariant index of a Dirac operator on the free loop space $Map(S^1,M)$. It is also an elliptic genus, which Ochanine describes much better than I could here.

This is supposed to have especially interesting behavior on certain manifolds. An orientation of a manifold is a lift of the structure of its tangent bundle from the orthogonal group $O(n)$ to the special orthogonal group $SO(n)$, which can be regarded as choosing data that exhibits triviality of the first Stiefel-Whitney class $w_1(M)$. A Spin manifold has its structure group further lifted to $Spin(n)$, trivializing $w_2(M)$. For Spin manifolds, the first Pontrjagin class $p_1(M)$ is canonically twice another class, which we sometimes call "$p_1(M)/2$"; a String manifold has a lift to the String group trivializing this class. Just as the $\hat A$-genus is supposed to take integer values on manifolds with a spin structure, it was argued by Witten that the Witten genus of a String manifold should take values in a certain subring: namely, power series in $\Bbb{Z}[[q]]$ which are modular forms. This is a very particular subring $MF_*$ isomorphic to $\Bbb{Z}[c_4,c_6,\Delta]/(c_4^3 - c_6^2 - 1728\Delta)$.

The development of the universal elliptic cohomology theory ${\cal Ell}$, its refinement at the primes $2$ and $3$ to topoogical modular forms $tmf$, and the so-called sigma orientation were initiated by the desire to prove these results. They produced a factorization of the Witten genus $MString_* \to \Bbb{C}[[q]]$ as follows: $$ MString_* \to \pi_* tmf \to MF_* \subset \Bbb{C}[[q]] $$ Moreover, the map $\pi_* tmf \to MF_*$ can be viewed as an edge morphism in a spectral sequence. There are also multiplicative structures in this story: the genus $MString_* \to \pi_* tmf$ preserves something a little stronger than the multiplicative structure, such as certain secondary products of String manifolds and geometric "power" constructions.

What does this refinement give us, purely from the point of view of manifold invariants?

  • The map $\pi_* tmf \to MF_*$ is a rational isomorphism, but not a surjection. As a result, there are certain values that the Witten genus does not take, just as the $\hat A$--genus of a Spin manifold of dimension congruent to 4 mod 8 must be an even integer (which implies Rokhlin's theorem). Some examples: $c_6$ is not in the image but $2c_6$ is, which forces the Witten genus of 12-dimensional String manifolds to have even integers in their power series expansion; similarly $\Delta$ is not in the image, but $24\Delta$ and $\Delta^{24}$ both are. (The full image takes more work to describe.)

  • The map $\pi_* tmf \to MF_*$ is also not an injection; there are many torsion classes and classes in odd degrees which are annihilated. These actually provide bordism invariants of String manifolds that aren't actually detected by the Witten genus, but are morally connected in some sense because they can be described cohomologically via universal congruences of elliptic genera. For example, the framed manifolds $S^1$ and $S^3$ are detected, and Mike Hopkins' ICM address that Drew linked to describes how a really surprising range of framed manifolds is detected perfectly by $\pi_* tmf$.

These results could be regarded as "the next version" of the same story for the relationship between the $\hat A$-genus and the Atiyah-Bott-Shapiro orientation for Spin manifolds. They suggest further stages. And the existence, the tools for construction, and the perspective they bring into the subject have been highly influential within homotopy theory, for entirely different reasons.

Hope this provides at least a little motivation.

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TMF has been used to solve classical topological problems. For example Bruner, Davis and Mahowald obtained new results regarding nonimmersions of real projective spaces in Euclidean space (http://hopf.math.purdue.edu//Bruner-Davis-Mahowald/eo2.pdf).

One very exciting sounding paper (not yet available) is by Behrens, Hopkins, Hill and Mahowald:

We completely determine the image of the Hurewicz homomorphism for tmf. We draw certain conclusions on which dimensions can contain exotic spheres. In particular, the only dimensions below 126 which contain no exotic spheres are 1,2,3,5, 6, 12, 61, and perhaps 4.

Some slides for a talk about this is available here: http://math.mit.edu/~mbehrens/presentations/Exotic_spheres.pdf

I don't know if it falls into classical topology, but the calculations of tmf have been used to find minimal $v_2$-periodic self maps (for example http://math.mit.edu/~mbehrens/papers/v2_32.pdf).

I know you asked about topological problems, but it seems remiss to not points out that there are also connections to number theory. Perhaps a good starting point is the ICM address of Hopkins: http://arxiv.org/pdf/math/0212397v1.pdf

From the mathscinet review:

The article describes how the study of tmf relates the Hopf fibration to the Weierstrass ℘-function, and leads to a proof of a theorem of Borcherds on congruences for modular forms arising from $\theta$-functions associated to lattices. It discusses the relationship between tmf and the theory of p-adic forms of Serre and Katz. It explains that tmf is the natural receptacle for the Witten genus, hinting at an intimate and still incompletely understood relationship to string theory.

Another paper I'm fond of is a paper of Mark Behrens: http://math.mit.edu/~mbehrens/papers/betagt.pdf He associates to each additive generator of the 2 line of the Adams-Novikov spectral sequence a certain modular form. This is akin to the connection between the Image of J and Bernoulli numbers.

I'm sure I'm just scratching the surface here.

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