is it permissible to inflate a ball on shabbos
There is controversy about this, but the poskim say that if it is the first time the ball is being blown up that it is not permitted because it is makeh b’patish, as it is now completing it, and making it ready for use. Additionally, if the ball (or balloon) needs to be tied, then even if was already blown up once, it is not permitted ,since the person may come to tie it on shabbos.
Shmiras Shabbos Khilchoso 16-7 in the name of R’ S. Z. Auerbach zt”l, Yesodei Yisurun 4- pg. 270, Zachor Vshamor- Makeh B’patish, Sefer 39 Melachos pg. 1156. Also see Minchas Yitchok 6-30, Chelkas Yackov 3-159.