Opening an umbrella on Shabbat is generally considered to be forbidden because of building. The linked article notes that while the g'mara (not specifically cited) does permit opening a folding chair, even though that creates a "tent" over the ground below, the purpose of opening the chair isn't to create the tent, while the purpose of erecting a structure above oneself is:
Accordingly, the Noda Be-Yehuda (work of responsa by Rabbi Yechezkel Landau of Prague, 1713-1793), among others, forbade opening umbrellas on Shabbat. This is the position taken by the vast majority of recent and contemporary authorities, as well, including Rabbi Moshe Halevi, in his work Menuchat Ahava, and Chacham Ovadia Yosef, in his work Yechaveh Da'at. It should be noted that once Halacha forbids opening an umbrella, closing an umbrella becomes forbidden, as well.
In addition, this article notes that
a chair folds and unfolds directly into the desired position. An umbrella, however, must be fastened or locked into place in order to assume the desired position. (Biur Halacha 315:7)