Atiyah–Jones conjecture

In mathematics, the Atiyah–Jones conjecture is a conjecture about the homology of the moduli spaces of instantons. The original form of the conjecture considered instantons over a 4 dimensional sphere. It was introduced by Michael Francis Atiyah and John D. S. Jones (1978) and proved by Charles P. Boyer, Jacques C. Hurtubise, and Benjamin M. Mann et al. (1992, 1993). The more general version of the Atiyah–Jones conjecture is a question about the homology of the moduli spaces of instantons on any 4 dimensional real manifold, or on a complex surface. The Atiyah–Jones conjecture has been proved for Ruled Surfaces by R. J. Milgram and J. Hurtubise, and for Rational Surfaces by Elizabeth Gasparim. The conjecture remains unproved for other types of 4 manifolds.

References

• Atiyah, Michael Francis; Jones, John D. S. (1978), "Topological aspects of Yang-Mills theory", Communications in Mathematical Physics, 61 (2): 97–118, Bibcode:1978CMaPh..61...97A, doi:10.1007/bf01609489, ISSN 0010-3616, MR 0503187
• Boyer, Charles P.; Hurtubise, Jacques C.; Mann, Benjamin M.; Milgram, R. James (1992), "The Atiyah–Jones conjecture", Bulletin of the American Mathematical Society, New Series, 26 (2): 317–321, arXiv:math/9204226, doi:10.1090/S0273-0979-1992-00286-0, ISSN 0002-9904, MR 1130447
• Boyer, Charles P.; Hurtubise, Jacques C.; Mann, Benjamin M.; Milgram, R. James (1993), "The topology of instanton moduli spaces. I. The Atiyah–Jones conjecture", Annals of Mathematics, Second Series, 137 (3): 561–609, doi:10.2307/2946532, ISSN 0003-486X, JSTOR 2946532, MR 1217348
• Hurtubise, J.; Milgram, R.J.; The Atiyah-Jones conjecture for ruled surfaces. Journal für die reine und angewandte Mathematik 466 (1995): 111-144.
• Gasparim, Elizabeth; The Atiyah-Jones conjecture for rational surfaces. Advances in Mathematics 218, (4) (2008) 1027-1050, https://doi.org/10.1016/j.aim.2008.03.004.