## A question about the topological proofs of Bott periodicity

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).

1. Source and target of the Bott map $\beta: BU \to \Omega SU$ are homotopy commutative $H$-spaces and $\beta$ is an $H$-map.
2. Therefore, $H_* (BU)$ and $H_* (\Omega SU)$ are commutative rings under the Pontrjagin product and $\beta$ induces a ring homomorphism.
3. 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 $1$).
4. $\beta \circ \lambda: \mathbb{CP}^{\infty} \to \Omega SU$ 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.
5. Anything proven so far implies that $\beta$ is surjective in homology.
6. 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.
7. 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?:

1. prove the cup product structure on $H^* (BU)$ (I wish to do this anyway)
2. compute the additive structure of $H^* (\Omega SU)$.
3. Identify some concrete cohomology classes of $\Omega SU$, for example given as transgressions of classes of $SU$ or double transgressions of classes of $BSU$.
4. 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.

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How are you computing $H_*(BU)$ and $H_*(SU)$? If you are using the Serre SS isn't the differential still a derivation? or is this still not enough to tell you about the Pontrjagin product? Great question by the way. – Sean Tilson Jan 28 2011 at 2:56

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