Benjamin M. Mann

The topology of instanton moduli spaces. I: The Atiyah-Jones conjecture

In this paper we study the global geometry and topology of the moduli spaces of based SU(2)-instantons over the 4-sphere S4 . These instanton moduli spaces have a rich history and have been analyzed from many points of view. Originally these spaces, which we denote by Mk, were defined as solution spaces (modulo gauge equivalence) to certain partial differential equations, namely the self-duality equations associated to the Yang-Mills functional in SU(2) gauge theory. They have been successfully studied from this point of view by Taubes ([T1], [T2]), Uhlenbeck [U] and others. An important alternative approach was initiated by Ward [W], who related instantons to certain holomorphic bundles on CP3, and was continued by Atiyah and Ward in [AW]. This allowed the classification of instantons on

Hypercomplex structures on Stiefel manifolds

4 in terms of quaternionic linear algebra by Atiyah, Drinfeld, Hitchin and Manin [ADHM]. This holomorphic approach was further extended by Donaldson [D], who showed that these bundles were determined by their restriction to a Cp2 and that the restricted bundles only had to satisfy the constraint of being trivial on a fixed line in C2R2. Hurtubise [Hul] then exploited this fact to study the moduli spaces Mk, as did Atiyah [A] to show that Mk arise naturally in the theory of holomorphic maps into loop groups; this latter approach was continued by Gravesen [G]. Atiyah and Jones [AJ] obtained the first results and formulated the foundational questions on the global topology of these moduli spaces. Recall that an element of Mk is a based gauge-equivalence class of a connection on the principal SU(2) bundle over

Algebraic cycles and infinite loop spaces

4, denoted by Pk with second Chern class k, satisfying the self-duality equations. There is a natural forgetful map to the based equivalence classes of all connections in Pk. Atiyah and Jones [AJ] showed that this target space, which we denote by Bk, is homotopy equivalent

The Atiyah-Jones conjecture

This paper describes a family of hypercomplex structures {% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFqessaaa!4076!\[\mathcal{I}\]a(p)}a=1,2,3 depending on n real non-zero parameters p = (p1,...,pn) on the Stiefel manifold of complex 2-planes in ℂn for all n > 2. Generally, these hypercomplex structures are inhomogenous with the exception of the case when all the pis are equal. We also determine the Lie algebra of infinitesimal hypercomplex automorphisms for each structure. Furthermore, we solve the equivalence problem for the hypercomplex structures in the case that the components of p are pairwise commensurable. Finally, some of these examples admit discrete hypercomplex quotients whose topology we also analyze.

The geometry and topology of 3-Sasakian manifolds.

SummaryIn this paper we use recent results about the topology of Chow varieties to answer an open question in infinite loop space theory. That is, we construct an infinite loop space structure on a certain product of Eilenberg-MacLane spaces so that the total Chern map is an infinite loop map. An analogous result for the total Stiefel-Whitney map is also proved. Further results on the structure of stabilized spaces of alebraic cycles are obtained and computational consequences are also outlined.

3-Sasakian manifolds

The purpose of this note is to announce our proof of the Atiyah-Jones conjecture concerning the topology of the moduli spaces of based SU(2) instantons over S 4 . Full details and proofs appear in our paper [BHMM1]