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The Atiyah-Jones conjecture for classical groups and Bott periodicity. (English) Zbl 0870.58012
An $$L$$-stratification is given of $${\mathcal M} (G)_k$$, the based moduli space of instantons of charge $$k$$ over $$S^4$$ with structure group $$G$$ being $$SO(n)$$ or $$Sp(n)$$. The main result obtained is as follows:
For $$G=SO(n)$$ with $$n>6$$ or $$G=Sp(n)$$ and for all $$k> 0$$ and all primes $$p$$, the Taubes inclusion map $$\iota_k: {\mathcal M}_k(G) \to{\mathcal M}_{k+1} (G)$$ induces an isomorphism in homology $(\iota_k)_t: H_t\bigl({\mathcal M}_k (G); \mathbb{A}\bigr) \cong H_t \bigl({\mathcal M}_{k+1} (G); \mathbb{A}\bigr)$ for $$t\leq q=q(k)= [k/2]-1$$ and $$\mathbb{A}= \mathbb{Z}$$ or $$\mathbb{Z}/p$$.
The stratifications also lead naturally to the computation of the fundamental groups. It is shown that $${\mathcal M}_k (SO(n))$$ are all simply connected for $$n>6$$, and that the fundamental groups of $${\mathcal M} (Sp(n))$$ are always $$\mathbb{Z}/2$$. The argument of Boyer-Hurtubise-Mann-Milgram is applied to confirm the Atiyah-Jones conjecture in these cases. To be precise, it is shown that:
For all positive integers $$k$$, the map $(\vartheta_k)_t: \pi_t \bigl({\mathcal M}_k (G) \bigr) \to\pi_t (\Omega^3_0G)$ on homotopy groups induced from the map $$\vartheta_k: {\mathcal M}_k(G)) \to\Omega^3_0G$$ is an isomorphism for $$t\leq q(k)= [k/2]-1$$ if $$G=SO(n)$$, $$n>6$$; and for $$t\leq q(k)= [k/2]-2$$ if $$G=Sp(n)$$.
Following Kirwan, the best possible ranges $$q(k)$$ are given for all classical groups except for a few possible cases when the sizes of the groups are small. Using a simple argument of Sanders, the stabilization results as the ranks of the groups go to infinity are obtained. (The stabilization of this sort was firstly studied by Kirwan.) Combination of these results with the Atiyah-Jones conjecture proved for classical groups results Bott periodicity.
Reviewer: Y.L.Tian (New Yor)

##### MSC:
 58D27 Moduli problems for differential geometric structures 55Q52 Homotopy groups of special spaces 57T99 Homology and homotopy of topological groups and related structures
##### Keywords:
instanton moduli; Atiyah-Jones conjecture; Bott periodicity
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