It’s finally time we shed some light on the technicalities behind this powerful tool. It took me some time digesting all these materials and I have to thank all the people who helped me understanding these passages. We will identify the differentials of the AHSS for spin-bordism (since it’s the one I need for my thesis), but the method we are going to use can be generalised to any AHSS, provided we have enough informations about the cohomology group of the spectra involved.
Since we will make an extensive use of this tool, it’s important that we clarify some technicalities about it. Our first aim is to prove the following lemma, which can be found as Lemma 2.3.2. page 27 in [Tei92]
Let be a spectrum and the Atiyah-Hirzebruch spectral sequence as above.
- The differential is the dual of
- The differential is reduction mod composed with the dual of .
We start with a definition:
Definition 1. An homology operation is a natural transformation between homology functors.
We call them stable if they commute with the suspension isomorphism, in analogy with the well-known stable cohomology operations. Now we want to relate this operations with the more famous cohomology operations, and we will do it as follows:
Lemma 2. Let be two spectra. Let be the respective homology and cohomology theories . There is a bijection between stable homology operations and stable cohomology operations for .
Proof. By Yoneda lemma, we know that stable cohomology operations are represented by (homotopy class of) maps of spectra . Therefore given , we can define the following natural transformation:
Since we used a map of spectra, the natural transformation preserves the suspension isomorphism (See [Rud] page 69) and it represents a stable homology operation by definition. Now let be a stable homology operation. Since is determined by its value on finite spectra (just use a colimit argument), it’s enough to consider just the case where is a finite CW-spectrum. We can apply the Spanier-Whitehead duality to pass from cohomology to homology, and we obtain:
Therefore we created a cohomology operation, and after checking carefully the construction of such duality, it’s clear that these two assignations are one inverse of the other.
Now we want to give a construction of the AHSS in such a way that the differentials are clearly induced by maps of spectra. This is done in the paper [Mau] of Maunder, where the author build a version of the AHSS via an exact couple given by the Postnikov tower of the (co)homology theory. It’s clear that the construction done there (for spaces) can be lifted to the setting of spectra. He then identifies the differentials with -invariants of the spectrum representing the (co)homology theory. We can say something similar for the edge homomorphisms too, in fact they are defined from the map in the definition of the exact couple. By the work of Maunder, all of these maps respect the suspension isomorphism since they come from maps of spectra (See here for a long argument or just use Yoneda Lemma).
Recall this result.
Theorem. Serre’s Theorem on the cohomology of . For we quote the results obtained by Serre in [Ser]. In each of the following, we describe the algebras over the Stennrod algebra.
- Let be the mod reduction of the fundamental class,
We are ready now to prove Lemma 1
Proof. Using the inclusion of the bottom cell , which induces an isomorphism in homotopy in degree , we can compute the differentials for the spectral sequence:
In fact we are interested in differentials which start at most at the second row, since the third is trivial thanks to . Now the differentials are stable homology operations and thus we have the following cases:
- . In this case, the homology operation is classified by elements of , where is the Eilenberg-MacLane spectrum which represents singular homology with coefficients . By definition, . In fact the Steenrod algebra is well understood, and we know that the cohomology group we are interested in is generated by the Steenrod square . Now recall the identification:, where denotes the sphere spectrum. We can consider the following commutative diagram given by the Spanier-Whitehead Duality: Where is the map representing the stable homology operation . Now notice that we have the pairing given by where is the duality and the multiplication given by the ring spectrum structure. This is (under the duality isomorphism) the well-known Kronecker pairing which in the case of field coefficient is a perfect pairing. For this reason we can identify with the dual of , and becomes the dual of a stable cohomology operation . After plugging in some test-space (as done in [Tei92]), one realises that it’s not the trivial operation, therefore the claim.
- . Here the reasoning is a little bit more involved. First of all, one has to realise that now we are interested in elements of , which by definition is . Therefore we need to compute this last cohomology group. Using the isomorphism we need to figure out how does the look like for big enough . By Serre’s Theorem , we have that , where is the fundamental class. It’s easy to see that the colimit is given by the equivalence class of this element, therefore we have only two possible homotopy classes of map of spectra, the trivial one, and the one induced by . As above, by a careful choice of test space, one can see that can’t be always trivial, therefore it remains to see what’s the effect of on our homology groups in order to be able to compute it explicitly. To this end, we consider the following diagram: Now notice that . By definition, an element can be represented by a map . The effect of composition with it’s the well-known reduction modulo in homology. After that, by the same reasoning above, the effect of composition with is given by the dual of the Steenrod square on the cohomology of .
We can give a nice description of the horizontal edge homomorphism in the AHSS for Oriented and Spin Bordism. Actually this can be generalised to other bordism theories, but we are interested in these two.
Definition 2. Let be or . We define the Steenrod-Thom map for -bordism as follows:
This map is clearly natural.
We start by recalling the geometric interpretation of the -bordism homology in the relative case.
Lemma 3. Let be a CW-pair. We can identify with the abelian group of singular manifolds (possibly with boundary) with spin structure, i.e. it’s elements are -bordism classes of manifolds with maps . The boundary operator turns out to be equal to the map .
Proof. See the observation in [Rud] page 289.
The Steenrod-Thom map can be easily generalized to a map as follows:
Notice that the following diagram commutes
Since . Therefore the family of maps represents an homology operation and by Lemma 2 is induced by a morphism of spectra .
We are now ready to prove the second main result (Prop 7.23 page 292 in [Rud])
Proposition. The edge homomorphism:
is given by the Steenrod-Thom homomorphism.
Proof. First of all, notice that the edge homomorphism is a stable homology operation. In fact if we use the exact couple given in [Mau] to define the AHSS, every map involved is clearly a stable homology operation. Therefore it has to be represented by an element in , where is the Thom class of the spectrum . Moreover, similarly to what we did for the differential, we see that the edge homomorphism acts by post-composition of for some :
Since clearly generates such cohomology group, by Proposition 5.24.i page 260 in [Rud] we have that is an isomorphism. We choose such that . After plugging into the AHSS some test spaces, it’s clear that the edge homomorphism is represented by and not by any of its multiples. Now it remains to be shown that has the same effect on homology as the Steenrod-Thom map. To this end, consider the map of spectra . Since is an isomorphism the element must be a generator. So . But both generators maps and therefore
[Tei92], P. Teichner, Topological four-Manifolds with Finite Fundamental Group, PhD thesis
[Rud], Rudyak, On Thom spectra, Orientability, and Cobordism, Corrected printing 2008, Springer Monographs in Mathematics
[Mau], C.R.F. Maunder, The spectral sequence of an extraordinary cohomology theory, Mathematical Proceedings of the Cambridge Philosophical Society 59, 1963
[Ser], J.-P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg-Maclane, Comment. Math. Helv. 27 (1953), 198-232.