Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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)?

share|improve this question
1  
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

1 Answer 1

up vote 5 down vote accepted

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 Milgram, who confirmed that there are no further (significant) results.

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.