# Conjecture on homotopy groups of moduli space of self-dual connections

In the paper Stability in Yang-Mills Theories (1983), Taubes puts a (topological) bound on the Hessian of the YM-action on $S^4$. He consequently conjectured:
"The inclusion $\mathcal{M}_n\hookrightarrow\mathcal{B}_n$ induces an isomorphism of the pointed homotopy groups $\pi_k(\cdot)$ for $k\le 2|n|$."
Here $\mathcal{M}_n$ is the moduli space of self-dual connections on a principal $SU(2)$-bundle over $S^4$ of degree $n\in\mathbb{Z}$, and $\mathcal{B}_n$ is the space of pairs $(P,A)$ where $P$ is a principal $SU(2)$-bundle over $S^4$ satisfying $n=-c_2(P\times_{SU(2)}\mathbb{C}^2)$ (instanton number) and $A$ is a smooth connection on $P$.

This was 30 years ago, at a time when Yang-Mills was being squeezed for useful information. I want to guess this has been proven by now, but I cannot immediately locate a proof.
Is this still an open problem?
If not, where is the proof? If so, what is currently known about the problem (with references)?

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This is very close to the Atiyah-Jones conjecture, proved by Boyer-Hurtubise-Mann-Milgram (The topology of instanton moduli spaces. I. The Atiyah-Jones conjecture. Ann. of Math. (2) 137 (1993), no. 3, 561–609.) You might start with this paper. –  Danny Ruberman Jun 25 '13 at 20:12

On afterthought I now asked Taubes himself, who pointed me to the paper The Topology of Instanton Moduli Spaces (Boyer-Hurtubise-Mann-Milgram). This conjecture is a form of the Atiyah-Jones Conjecture. The paper is essentially what is known about this problem, and in particular the above conjecture is still open. Atiyah-Jones showed that $B_n\simeq \Omega^3_nSU(2)$, and this paper proves the partial result:
The (forgetful) inclusion map $\mathcal{M}_n\to \Omega^3_nSU(2)$ is a homotopy equivalence through dimension $\lfloor\frac{n}{2}\rfloor-2$.
I also asked Hurtubise, who remembers improving the stability range from $\frac{n}{2}$ to $n$ by computing a spectral sequence differential, but cannot remember the actual reference.