There are purely topological proof of Botts periodicity theorem, the first one given by Dyer and Lashof. I am heading to discuss the proof
in my lecture course on homotopy theory (as a final chord and as an application of the Leray-Serre spectral sequence).
Textbook references that I have consulted are: Dyer "Cohomology theories", Mimura-Toda: "Topology of Lie groups", Switzer "Algebraic topology".
I also want to advertise a nice paper by Kono and Tokunaga which uses methods are well beyond the scope of my course.
There are several versions of the argument, but the idea is always the same. Here is a sketch (in the complex case; a similar argument works in the real
case, but it is at least six times as complicated).
- Source and target of the Bott map $\beta: BU \to \Omega SU$ are homotopy commutative $H$-spaces and $\beta$ is an $H$-map.
- Therefore, $H_* (BU)$ and $H_* (\Omega SU)$ are commutative rings under the Pontrjagin product and $\beta$ induces a ring homomorphism.
- Consider the classifying map $\lambda :\mathbb{CP}^{\infty} \to BU$ of the Hopf
bundle. Tracing throgh the definition of the Bott map shows that $\beta \circ \lambda: \mathbb{CP}^{\infty} \to \Omega SU$ is the adjoint of
the following map $\Sigma \mathbb{CP}^{\infty} \to SU$: $(z,l)$ goes to the linear map that is multiplication by $z$ on $l$ and $1$ on the
complement (OK, you have to normalize to make the determinant $1$). - $\beta \circ \lambda: \mathbb{CP}^{\infty} \to \Omega SU$ is injective in homology and the induced map $P [H_* (\mathbb{CP}^{\infty})] \to H_{*}(\Omega SU)$
($P$ means polynomial algebra) is a ring isomorphism. This is the part that makes heavy use of spectral sequences. - Anything proven so far implies that $\beta$ is surjective in homology.
- Since the homology groups of $BU$ and $\Omega SU$ are free abelian of the same rank (here on needs to know the additive structure of $H_* (BU)$),
$\beta$ is also injective in homology. - By the Hurewicz theorem, $\beta$ is a weak homotopy equivalence.
Version 1 of my question: Is it possible to modify the argument to use cohomology instead? Or is there a reason why the following sketch does
not work?:
- prove the cup product structure on $H^* (BU)$ (I wish to do this anyway)
- compute the additive structure of $H^* (\Omega SU)$.
- Identify some concrete cohomology classes of $\Omega SU$, for example given as transgressions of classes of $SU$ or
double transgressions of classes
of $BSU$. - Prove that the images of these classes under $\beta$ generate $H^* (BU)$.
This would have, in my opinion, the advantage that it uses the much more familiar cup product structure on $H^* (BU)$ instead of the Pontrjagin product structure on $H_* (\Omega SU)$. Therefore the proof is at least psychologically simpler.
A more technical and more focussed version of my question is:
Version 2: what are the inverse images $(\beta^*)^{-1}(c_k) \in H^{2k} (\Omega SU)$. I am looking for a formula that is explicit enough to show
that $\beta^*$ is surjective.
EDIT: one obvious answer could be that my sketch does not need the H-space structures and will fail for that reason.
Best Answer
There is a beautiful observation of my advisor John Moore that to my mind ought to be part of the focus of any such argument: the Hopf algebra $H_{\ast}(BU;Z)$, or equivalently $H^*(BU;Z)$ is self-dual. This is pure algebra and streamlines the presentation. My preferred presentation is in Section 21.6 of May and Ponto, More Concise Algebraic Topology, available at Amazon or University of Chicago Press. Incidentally, there is a proof of real Bott periodicity along the same lines due to Moore (and written up by Cartan if memory serves) in one of the 1950's Cartan seminars and another such proof by Dyer and Lashof. The funny thing is that each of these proofs finds one of the six equivalences one has to prove more difficult than all of the others, but they find difficulty with different ones: combining the easier of the respective arguments gives a clean and instructive homological proof.