Math 549. Elliptic
cohomology . Fall, 2003.
Lecture notes.
This page will
give a summary of each lecture as the semester progresses.
November 18 and 20
Continued the discussion of the 2-primary calculation for tmf. The approach I used is very
similar to that of Tilman Bauer in
his preprint "Computation of the homotopy of the spectrum tmf."
November 13
Began
the 2-primary calculation for tmf. The change-rings-isomorphism (A1.3.12 of the green
book) leads us to computing Ext groups for various modules over
A(1), the subalgebra of the
mod 2 Steenrod algebra generated by Sq1 and Sq2. Again we refer the reader to Rezk 's notes (dvi,
pdf)
for more details.
October 30
Finished
the 3-primary calculation for tmf. This entails the use of Toda backets (7.5.4 of the green
book) and Massey products (A1.4). The best reference for this is
the notes for a course given by Rezk at Northwestern in the
Spring of 2001 on topological modular forms (dvi,
pdf).
October 28
First
meeting. Discussed groupoids and Hopf algebroids; see the beginning of
section A1.1 of the green
book. The self homology of a ring spectrum E satisfying certain hypotheses
(2.2.5) is a Hopf algebroid. I described it explicitly for E=MU in terms of formal group laws;
see Theorems 4.1.11 and A2.1.16. Defined the cobar complex (A1.2.11)
for computing Ext over a Hopf algebroid, which is the E2-term of the Adams spectral sequence.
Second meeting. Introduced the Hopkins-Mahowald Hopf algebroid for elliptic curves, described in page 2 of my Newton talk. Its Ext group is the E2-term of the MU-based Adams spectral sequence converging to the homotopy of the spectrum tmf. The main tool here is the change-of-rings isomorphism of A1.3.12.
I used it to show that for any prime larger than 3, the spectral sequence is concentrated on the 0-line, showing that the homotopy of tmf is a polynomial algebra on two generators, in dimensions 8 and 12.
For the prime 3 the situation is more complicated. I calculated the Ext group, i.e., the Adams E2-term. This is related to a calcalation over the subalgebra P(0) of the mod 3 Steenrod algebra. There are differentials in the Adams spectral sequnce, that I will discuss next time.
Second meeting. Introduced the Hopkins-Mahowald Hopf algebroid for elliptic curves, described in page 2 of my Newton talk. Its Ext group is the E2-term of the MU-based Adams spectral sequence converging to the homotopy of the spectrum tmf. The main tool here is the change-of-rings isomorphism of A1.3.12.
I used it to show that for any prime larger than 3, the spectral sequence is concentrated on the 0-line, showing that the homotopy of tmf is a polynomial algebra on two generators, in dimensions 8 and 12.
For the prime 3 the situation is more complicated. I calculated the Ext group, i.e., the Adams E2-term. This is related to a calcalation over the subalgebra P(0) of the mod 3 Steenrod algebra. There are differentials in the Adams spectral sequnce, that I will discuss next time.
October 23
Discussed
the Adams spectral sequence for a generalized homology
theory E*. Under certain
assumptions on E (spelled out in section 2.2
of the green
book), its E2-term can be
identified as Ext over the Hopf algebroid E*(E). A Hopf
algebroid is a cogroupoid object in the category of commutative
algebras over a suitable ground ring. Hopf algebroids are discussed at
length in Appendix 1 of the green
book.
October 21
First
meeting. Discussed Serre's method of computing homotopy groups. This
means choosing a map from the simply connected space X to an Eilenberg-Mac Lane space
K that induces an isomorphism on the first
nontrivial homotopy group. Then the fiber X' has the same homotpygroups as X in higher dimensions. Knowing
the first nontrivial homology group of X' (which we can compoute using
the Serre spectral sequence) tells us what its first nontrivial
homotopy group is. In the stable range the Serre spectral sequence
reduces to a long exact sequence. In any case we need to know the
cohomology of all relevant Eilenberg-Mac Lane spaces, which leads to a
discussion of the mod p Steenrod
algebra A.
Serre's method was used only for a few years because it was supplanted by the Adams spectral sequence in 1959. In it we use maps to wedges of mod p Eilenberg-Mac Lane spaces which induce surjections in cohomology rather than isomorphisms in bottom homotopy groups. Geometrically this leads to a diagram called an Adams resolution and algebraically it leads to a free A-resolution of H*(X).
Second meeting. Explained how the free A-resolution of H*(X) leads one to identify Ext over A as the E2-term of the Adams spectral sequence.
Then I reformulated the Adams spectral sequence in homological (rather than cohomological) terms. Why do this? For the classical Adams spectral sequence the two formulations are formally equivalant, and the cohomological one may be easier because it is more natural to think about modules over the Steenrod algebra than about comodules over its dual. However when one considers that Adams spectral sequence based on a generalized homology theory such as MU*, the cohomological approach entails substantial technical headaches which can be completely avoided by using the homological approach.
Serre's method was used only for a few years because it was supplanted by the Adams spectral sequence in 1959. In it we use maps to wedges of mod p Eilenberg-Mac Lane spaces which induce surjections in cohomology rather than isomorphisms in bottom homotopy groups. Geometrically this leads to a diagram called an Adams resolution and algebraically it leads to a free A-resolution of H*(X).
Second meeting. Explained how the free A-resolution of H*(X) leads one to identify Ext over A as the E2-term of the Adams spectral sequence.
Then I reformulated the Adams spectral sequence in homological (rather than cohomological) terms. Why do this? For the classical Adams spectral sequence the two formulations are formally equivalant, and the cohomological one may be easier because it is more natural to think about modules over the Steenrod algebra than about comodules over its dual. However when one considers that Adams spectral sequence based on a generalized homology theory such as MU*, the cohomological approach entails substantial technical headaches which can be completely avoided by using the homological approach.
October 9
Discussed
several examples of the Serre spectral sequence:
- The fibering of an odd dimensional sphere over a complex projective space. Here the fiber is S1, and the sepctral sequence reduces to a long exact sequence called the Gysin sequence. IN this case we know the answer in advance.
- The fibering of a finite dimensional unitary group over an odd dimensional sphere. Here the base is a sphere and the sepctral sequence reduces to a long exact sequence called the Wang sequence.
- The path fibration over the Eilenberg Mac Lane space K(Z/2, 2). Here we know the cohomology of the fiber and the total space, and we use the spectral sequence to deduce the cohomology of the base, which turns out to be a certain polynomial algebra.
- The fibration leading to the 3-connected cover of S3, for whic the base is S3 and the fiber is K(Z, 2). This leads to a proof that the first p-torsion in the homotopy of S3 occurs in dimension 2p.
October 7
First
meeting. Continued the discussion of spectra, defining ring spectra
and generalized homology and cohomology. Mentioned the problem of
defining the smash product of spectra, referring to Elmendorf, Kriz,
Mandell, and May, Rings, modules, and algebras in stable
homotopy theory.
Second meeting. Explained how the universal formal group law shows up in the complex cobordism of infinite dimensional complex projective space. Defined complex oriented spectra. For more details see the first few chapters of Part II of Adams' blue book, Stable homotopy and generalised homology.
Began discussing the Serre spectral sequence.
Second meeting. Explained how the universal formal group law shows up in the complex cobordism of infinite dimensional complex projective space. Defined complex oriented spectra. For more details see the first few chapters of Part II of Adams' blue book, Stable homotopy and generalised homology.
Began discussing the Serre spectral sequence.
October 2
Gave the
complex analog of the cobordism isomorphism of the previous lecture.
This motivates the definition of a spectrum in the sense of homotopy,
which I discussed at length.
This material can be found the first 4 chapters of part III of Adams' Stable homotopy and generalised homology. The theory of spectra has recently been revolutionized by the work of Elmendorf, Kriz, Mandell, and May, Rings, modules, and algebras in stable homotopy theory.
This material can be found the first 4 chapters of part III of Adams' Stable homotopy and generalised homology. The theory of spectra has recently been revolutionized by the work of Elmendorf, Kriz, Mandell, and May, Rings, modules, and algebras in stable homotopy theory.
September 30
Finished
the proof of Mischenko's theorem by identifying the Milnor
hypersurface Hm,1. Defined Thom
spaces and gave some examples. Described the Pontrjagin-Thom
construction, which leads to a homomorphism from a certain cobordism
group to a homotpy group of a certain Thom space. The Thom
Transversality theorem (which I did not state) implies that this is an
isomorphism.
This material can be found in Chapters 17 and 18 of Milnor and Stasheff.
This material can be found in Chapters 17 and 18 of Milnor and Stasheff.
September 25
Computed the Chern numbers of 4-dimensional projective hypersurfaces.
Defined the Milnor hypersurfaces Hm,n and used them to
define the formal group over MU*. Computed its
logarithm modulo identifying the Hm,1.
September 23
Proved
a lemma (Lemma 4.4 of Milnor and
Stasheff) identifying the tangent bundle of complex projective
space which enables us to compute its Chern classes. Defined the
tensor product of two line bundles and discussed projective
hypersurfaces. Used this to compute the genus of a projective plane
curve of degree d.
In a geometric digression I showed that such a curve is homeomorphic to the boundary of a thickening of the complete graph on d vertices (i.e., the 1-skeleton of the standard (d-1)-simplex Δd-1) embedded in Euclidean 3-space. This can be generalized as follows. A smooth curve in CP1+n defined by equations of degrees d1, d2, ..., dn has a similar description in terms of the 1-skeleton of the corresponding product of standard simplices. As far as I know, this is not in the literature.
In a geometric digression I showed that such a curve is homeomorphic to the boundary of a thickening of the complete graph on d vertices (i.e., the 1-skeleton of the standard (d-1)-simplex Δd-1) embedded in Euclidean 3-space. This can be generalized as follows. A smooth curve in CP1+n defined by equations of degrees d1, d2, ..., dn has a similar description in terms of the 1-skeleton of the corresponding product of standard simplices. As far as I know, this is not in the literature.
September 18
Discussed characteristic classes of real and complex vector bundles.
They arise from the cohomology of the relevant classifying spaces and
can also be defined axiomatically. Defined Chern numbers and showed
that they are cobordism invariants.
All of this material can be found in Milnor and Stasheff and some of it is in Chapter 3 of Hatcher.
All of this material can be found in Milnor and Stasheff and some of it is in Chapter 3 of Hatcher.
September 16
Discussed real and complex
vector bundles. Examples cited were the trivial bundle, the
tautological line bundle over a projective space, the normal bundle of
a submanifold, and the tangent bundle of a manifold. Defined the
induced (by a continuous map) bundle and the Whitney sum of two bundles
over the same space. The restriction of the tangent bundle of an
ambient manifold to a submanifold is the Whitney sum of the
submanifold's tangent and normal bundles. For the purposes of
cobordism theory, a complex manifold is defined to be a manifold which
can be embedded in Euclidean space with a complex normal bundle.
Defined the Grassmannian of complex n-planes in (n+k)-space and the tautological n-plane bundle over it. Taking
the direct limit as k goes to infinity leads to the
classifying space
BU(n) for all n-plane bundles.
The standard reference for this material is the classic book Characteristic classes by Milnor and Stasheff. It is also covered online in the first chapter of Hatcher's Vector Bundles and K-Theory.
The standard reference for this material is the classic book Characteristic classes by Milnor and Stasheff. It is also covered online in the first chapter of Hatcher's Vector Bundles and K-Theory.
September 11
Discussed cobordism in the real
and
unoriented cases, describing the cobordism ring for each. The precise
definition of a "complex" manifold will be given later. Defined the
complex bordism groups of a space X and described the case where X is
a complex projective space. The complex cobordism ring MU*
is
isomorphic to the Lazard ring over which the universal FGL is
defined. Our goal for the next few lectures is to show that this
isomorphism corresponds to a geometrically defined FGL over MU*.
Some references for this and related material are Stong's Notes on cobordism theory, the first two parts of Adams' Stable homotopy and generalised homology, and the first section of Chapter 4 of my book Complex cobordism and stable homotopy theory.
Some references for this and related material are Stong's Notes on cobordism theory, the first two parts of Adams' Stable homotopy and generalised homology, and the first section of Chapter 4 of my book Complex cobordism and stable homotopy theory.
September 9
Discussed formal group laws.
Defined the
logarithm of a FGL in characteristic 0, the height in characteristic p,
and the universal FGL of Lazard. See the course home page for a link
to the FGL handout.
September 4
Discussed modular forms and formal group laws. Here are the transparencies I used, with
corrections.
September 2
Discussed elliptic curves as quotients of the complex numbers C
by a lattice Λ and as projective cubic curves.
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