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Abstract Let $${\mathrm{Diff}}_{0}(N)$$ represent the subgroup of diffeomorphisms that are homotopic to the identity. We show that if $$N$$ is a closed hyperbolic 4-manifold, then $$\pi _{0}{\mathrm{Diff}}_{0}(N)$$ is not finitely generated with similar results holding topologically. This proves in dimension-4 results previously known for $$n$$-dimensional hyperbolic manifolds of dimension $$n\ge 11$$ by Farrell and Jones in 1989 and $$n\ge 10$$ by Farrell and Ontaneda in 2010. Our proof relies on the technical result that $$\pi _{0}{\mathrm{Homeo}}(S^{1}\times D^{3})$$ is not finitely generated, which extends to the topological category smooth results of the authors. We also show that $$\pi _{n-4} {\mathrm{Homeo}}(S^{1} \times D^{n-1})$$ is not finitely generated for $$n \geq 4$$ and in particular $$\pi _{0}{\mathrm{Homeo}}(S^{1}\times D^{3})$$ is not finitely generated. These results are new for $n=4, 5$ and $$7$$. We also introduce higher dimensional barbell maps and establish some of their basic properties. 

We survey some recent geometric methods for studying Heegaard splittings of 3-manifolds.