Atiyah–Jones conjecture

{{Short description|Conjecture about the moduli space of instantons}}

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 {{harvs|txt | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Jones | first2=John D. S. | title=Topological aspects of Yang-Mills theory | url=http://projecteuclid.org/getRecord?id=euclid.cmp/1103904210 | mr=503187 | year=1978 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=61 | issue=2 | pages=97–118}} and proved by {{harvs|txt| last1=Boyer | first1=Charles P. | authorlink1=Charles P. Boyer | last2=Hurtubise | first2=Jacques C. |author2-link=Jacques Hurtubise (mathematician)| last3=Mann | first3=Benjamin M. | last4=Milgram | first4=R. James | year1=1992|year2=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

{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Jones | first2=John D. S. | title=Topological aspects of Yang-Mills theory | url=http://projecteuclid.org/getRecord?id=euclid.cmp/1103904210 | mr=503187 | year=1978 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=61 | issue=2 | pages=97–118 | doi=10.1007/bf01609489|bibcode = 1978CMaPh..61...97A | s2cid=122490773 }}
{{Citation | last1=Boyer | first1=Charles P. | last2=Hurtubise | first2=Jacques C.|author2-link=Jacques Hurtubise (mathematician) | last3=Mann | first3=Benjamin M. | last4=Milgram | first4=R. James | authorlink4=R. James Milgram | title=The Atiyah–Jones conjecture | doi=10.1090/S0273-0979-1992-00286-0 | mr=1130447 | year=1992 | journal=Bulletin of the American Mathematical Society |series=New Series | issn=0002-9904 | volume=26 | issue=2 | pages=317–321| arxiv=math/9204226 | s2cid=18497401 }}
{{Citation | last1=Boyer | first1=Charles P. | last2=Hurtubise | first2=Jacques C. |author2-link=Jacques Hurtubise (mathematician)| last3=Mann | first3=Benjamin M. | last4=Milgram | first4=R. James | title=The topology of instanton moduli spaces. I. The Atiyah–Jones conjecture | mr=1217348 | year=1993 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=137 | issue=3 | pages=561–609|jstor=2946532 | doi=10.2307/2946532}}
{{citation

| last1=Hurtubise | first1=J. C. |authorlink1=Jacques Hurtubise (mathematician)

| last2=Milgram | first2=R. J.

| title=The Atiyah-Jones conjecture for ruled surfaces

| journal=Journal für die reine und angewandte Mathematik

| volume=466

| date=1995

| pages=111–144

| doi=10.1515/crll.1995.466.111| s2cid=117414381 }}