In this paper we study the homology of a random Čech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.

a b s t r a c t G 2 -monopoles are solutions to gauge theoretical equations on noncompact 7-manifolds of G 2 holonomy. We shall study this equation on the 3 Bryant-Salamon manifolds. We construct examples of G 2 -monopoles on two of these manifolds, namely the total space of the bundle of anti-self-dual two forms over the S 4 and CP 2 . These are the first nontrivial examples of G 2 -monopoles.Associated with each monopole there is a parameter m ∈ R + , known as the mass of the monopole. We prove that under a symmetry assumption, for each given m ∈ R + there is a unique monopole with mass m. We also find explicit irreducible G 2 -instantons on Λ 2 − (S 4 ) and on Λ 2 − (CP 2 ).The third Bryant-Salamon G 2 -metric lives on the spinor bundle over the 3-sphere. In this case we produce a vanishing theorem for monopoles.

We construct the first nontrivial examples of Calabi-Yau monopoles. Our main interest on these, comes from Donaldson and Segal's suggestion [3] that it may be possible to define an invariant of certain noncompact Calabi-Yau manifolds from these gauge theoretical equations. We focus on the Stenzel metric on the cotangent bundle of the 3-sphere T * S 3 and study monopoles under a symmetry assumption. Our main result constructs the moduli of these symmetric monopoles and shows that these are parametrized by a positive real number known as the mass of the monopole. In other words, for each fixed mass we show that there is a unique monopole which is invariant in a precise sense. Moreover, we also study the large mass limit under which we give precise results on the bubbling behavior of our monopoles. Towards the end an irreducible SU (2) Hermitian-Yang-Mills connection on the Stenzel metric is constructed explicitly.

We initiate the systematic study of G 2 -instantons with SU(2) 2 -symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on R 4 × S 3 with its two explicitly known distinct holonomy G 2 metrics, which have different volume growths at infinity, exhibiting the different behaviour of instantons in these settings. We also give an explicit example of sequences of G 2 -instantons where "bubbling" and "removable singularity" phenomena occur in the limit.