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\subsection{Higgs physics in $\epem$ collisions at $\sqrt{s} = 500$~GeV}  The TLEP physics potential at this centre-of-mass energy would be similar to that of the linear colliders ILC and CLIC, which have nominal luminosities that are slightly lower comparable  at $\sqrt{s}=500$~GeV. The ILC TDR~\cite{cite:ILCTDR} shows that the addition of 500~$\infb$ at 500~GeV to the baseline programme with 250~$\infb$ at 250~GeV and 350~$\infb$ at 350 GeV would improve the precision on all Higgs boson couplings to light fermions and gauge bosons by less than a factor $\sqrt{2}$, still quite far from the sub-per-cent precision provided by TLEP at 240~GeV. The measurement of the invisible width of the Higgs boson would not be improved. On the other hand, the opening of the $\epemto \ttbar{\rm H}$ process allows the Htt coupling to be measured directly, typically with a precision of 14\% at ILC, and 10\% at TLEP. 10 to 15\%.  However, the improvement with respect to the TLEP measurement at the $\ttbar$ threshold, which has an accuracy of 13\%, is marginal. More importantly, these precisions are not competitive with the HL-LHC projections~\cite{1307.7135,1307.7292}. For example, the CMS collaboration would be able to measure the Htt coupling with an accuracy of 7\% using just the ${\rm H} \to \bbbar, \gamma\gamma$ decay channels and an integrated luminosity of 3~$\inab$. This precision could be improved to 5\% by combining the two LHC experiments, and to significantly better when the ${\rm H} \to \tau\tau$, ZZ, and WW leptonic decay channels are added. Similarly, the opening of the $\epemto {\rm ZHH}$ and $\nnbar{\rm HH}$ processes at $\sqrt{s} = 500$~GeV enables a ``measurement'' of the triple Higgs-boson self-coupling, $\lambda_{\rm H}$ with 50 to  80\% precision at ILC and 50\% precision at TLEP. precision.  Again, these accuracies are not competitive with the HL-LHC projections, for which a 30\% accuracy on $\lambda_{\rm H}$ is envisioned. At this stage of the study, it appears that once sufficient data are collected at 250 and 350 GeV, the potential gain in Higgs physics alone is not enough to justify an upgrade to a centre-of-mass of 500~GeV. (On the other hand, as discussed below, the appearance of some threshold for new physics above 350~GeV could change the picture entirely.)