Instanton symmetries and the entropy of compact manifolds

Many Euclidean Einstein manifolds possess continuous symmetry groups of at least one parameter and we consider here a classification scheme of $d$ dimensional compact manifolds based on the existence of such a one parameter group in terms of the fixed point sets of the isometries. We discuss applications of such a classification scheme, including the geometric interpretation of the entropy; there are intrinsic contributions to the entropy from the volumes of $(d-2)$ dimensional fixed point sets and contributions related to the cohomology structure of the orbit space of the isometry. We consider the relevance of such a decomposition of the entropy in the context of the no boundary proposal and cosmological processes, and generalise the discussion to compact solutions of gravity coupled to scalar and gauge fields.