Neuffer, D.V. Hart, T.L. Sayed, H. K. Summers, D.J. JACoW Geneva, Switzerland 978-3-95450-168-7 10.18429/JACoW-IPAC2015-TUBD2 English 1384-1386 TUBD2 emittance collider luminosity solenoid linac Contribution to a conference proceedings 2015 2015-06 https://doi.org/10.18429/JACoW-IPAC2015-TUBD2 http://jacow.org/ipac2015/papers/tubd2.pdf The final cooling system for a high-energy high-luminosity muon collider requires reduction of the transverse emittance by an order of magnitude to ~0.00003 m (rms, N), while allowing longitudinal emittance increase to ~0.1m. In the present baseline approach, this is obtained by transverse cooling of low-energy muons within a sequence of high field solenoids with low-frequency rf systems. Recent studies of such systems are presented. Since the final cooling steps are actually emittance exchange a variant form of that final system can be obtained by a round to flat transform in x-y, with transverse slicing of the enlarged flat transverse dimension followed by longitudinal recombination of the sliced bunchlets. Development of final exchange following lowest-emittance cooling is discussed.