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The well-rounded retract for $\mathrm{SL}_n(\mathbb{Z})$ is defined as theset of flat tori of unit volume and dimension $n$ whose systoles generate afinite-index subgroup in homology. This set forms an equivariant spine ofminimal dimension for the space of flat tori.  For both the Outer space $X_n$ of metric graphs of rank $n$ and theTeichm\"uller space $\mathcal{T}_g$ of closed hyperbolic surfaces of genus $g$,we show that the literal analogue of the well-rounded retract does not containan equivariant spine. We also prove that the sets of graphs whose systoles filleither topologically or geometrically (two analogues of a set proposed as aspine for $\mathcal{T}_g$ by Thurston) are spines for $X_n$ but that theirdimension is larger than the virtual cohomological dimension of$\mathrm{Out}(F_n)$ in general.

Failure of the well-rounded retract for Outer space and Teichm\"uller space