This paper contains a classification of smooth Kaluza-Klein reductions (by one-parameter subgroups) of the maximally supersymmetric anti de Sitter backgrounds of supergravity theories. We present a classification of one-parameter subgroups of isometries of anti de Sitter spaces, discuss the causal properties of their orbits on these manifolds, and discuss their action on the space of Killing spinors. We analyse the problem of which quotients admit a spin structure. We then apply these results to write down the list of smooth everywhere spacelike supersymmetric quotients of AdS₃ x S³(x ℝ⁴), AdS₄ x S⁷, AdS₅ x S⁵ and AdS₇ x S⁴, and the fraction of supersymmetry preserved by each quotient. The results are summarised in tables which should be useful on their own. The paper also includes a discussion of supersymmetry of singular quotients.

We consider branes $N=I \times \mathcal{S}_0$, where $\mathcal{S}_0$ is an $n$-dimensional space form, not necessarily compact, in a Schwarzschild-AdS$_{(n+2)}$ bulk $\mathcal{N}$. The branes have a big crunch singularity. If a brane is an ARW space, then, under certain conditions, there exists a smooth natural transition flow through the singularity to a reflected brane $\hat{N}$, which has a big bang singularity and which can be viewed as a brane in a reflected Schwarzschild-AdS$_{(n+2)}$ bulk $\hat{\mathcal{N}}$. The joint branes $N \cup \hat{N}$ can thus be naturally embedded in $\mathbb{R}^2 \times \mathcal{S}_0$, hence there exists a second possibility of defining a smooth transition from big crunch to big bang by requiring that $N \cup \hat{N}$ forms a $C^\infty$-hypersurface in $\mathbb{R}^2 \times \mathcal{S}_0$. This last notion of a smooth transition also applies to branes that are not ARW spaces, allowing a wide range of possible equations of state.

The topological part of the M-theory partition function was shown by Witten to be encoded in the index of an E₈ bundle in eleven dimensions. This partition function is, however, not automatically anomalyfree. We observe here that the vanishing W₇ = 0 of the Diaconescu-Moore-Witten anomaly [1] in IIA and compactified M-theory partition function is equivalent to orientability of spacetime with respect to (complex-oriented) elliptic cohomology. Motivated by this, we define an elliptic cohomology correction to the IIA partition function, and propose its relationship to interaction between 2- and 5-branes in the M-theory limit.