We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of $\operatorname{Lip}_1 (X, m)$, the set of all bounded $1$-Lipschitz functions $f : X \to \mathbb{R}^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $\mathcal{H}^n (f(S)) = 0$ is residual in $\operatorname{Lip}_1 (X, m)$. Conversely, if $E \subset X$ is $n$-rectifiable with $\mathcal{H}^n (E) \gt 0$, the set of all $f$ with $\mathcal{H}^n (f(E)) \gt 0$ is residual in $\operatorname{Lip}_1 (X, m)$.

These results provide a replacement for the Besicovitch–Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.

We study the rational homotopy types of classifying spaces of automorphism groups of smooth simply connected manifolds of dimension at least five. We give differential graded Lie algebra models for the homotopy automorphisms and the block diffeomorphisms of such manifolds.

Moreover, we use these models to calculate the rational cohomology of the classifying spaces of the homotopy automorphisms and block diffeomorphisms of the manifold ${\#}^g S^d \times S^d$ relative to an embedded disk as $g \to \infty$ The answer is expressed in terms of stable cohomology of arithmetic groups and invariant Lie algebra cohomology. Through an extension of Kontsevich’s work on graph complexes, we relate our results to the (unstable) homology of automorphisms of free groups with boundaries.