# Principal and fiber bundles as defined by Husemoller

In his book 'Fiber Bundles' Husemoller defines principal bundles and fiber bundles quite differently from how they are usually defined.

Specifically:

Definition: a right $G$-space $X$ is called effective if $G$ acts freely (or faithfully? It's ambiguous) on $X$ (that is, $xs = x$ implies $s = 1$).

Definition: Consider the space $X^* \subset X \times X$ given as $X^* = \{(x, xs) \mid x \in X, s \in G\}$. For all $(x, y) \in X^*$ there is a unique $\tau(x, y) \in G$ such that $x \tau(x, y) = y$. Thus we have the translation function $\tau: X^* \to G$. We call $X$ principal if $\tau$ is continuous.

Definition: A bundle $\xi: X \to B$ is called principal if $X$ is a principal $G$-space, and there exists a homeomorphism $f: X/G \to B$ inducing a bundle isomorphism $(1_X, f): \alpha(X) \to \xi$, where $\alpha$ is a functor from right $G$-spaces to bundles given by projection $X \to X/G$.

Definition: Consider a principal $G$-bundle $\xi: X \to B$, and let $F$ be a left $G$-space. The formula $(x, y)s = (xs, s^{-1}y)$ gives a structure of a right $G$-space on $X \times F$. Denote $X_F := (X \times F)/G$, and consider $\xi[F]: X_F \to B$ - a factorization of the composition $X \times F \to X \stackrel{\xi}{\to} B$ by the projection $X \times F \to X_F$, or, explicitly, $\xi[F] ((x,y)G) = \xi(x)$. Such a $\xi[F]$ is called the fiber bundle over $B$ with fiber $F$ and the associated principal bundle $\xi$. The group $G$ is called the structure group of the fiber bundle $\xi[F]$.

This is totally baffling to me, because the meaning of the terms is wildly different from the meaning in other sources I encountered so far, like Wikipedia; note that neither principal bundle nor fiber bundle is required to be locally trivial here! What are these things called in the mainstream literature? How do they arise, why are they interesting? I admit I have only started reading this section of Husemoller so I'll probably get the big picture if I continue reading, that's why I'm mostly interested in the terminology right now.

-
Well, Husemoller says "locally trivial" whenever he means locally trivial later on. –  t.b. Sep 10 '11 at 9:32
@Theo: Yeah, but what are the modern terms for what he calls 'principal bundles' and 'fiber bundles'? –  Alexei Averchenko Sep 10 '11 at 10:11

What Husemoller calls a principal bundle Richard S. Palais in his On the existence of slices for actions of non-compact Lie groups calls a Cartan principal bundle (introduced, to the best of my knowledge, by Jean-Pierre Serre in Seminar Cartan 1949-1950, perhaps incorrectly cited as 1948-1949), with the term principal bundle meaning any bundle induced by a free continuous group action.

Some facts about Cartan principal bundles:
1) Any locally trivial principal bundle is Cartan.
2) Any quotient space of a topological group by a closed subgroup is Cartan.
3) A Cartan principal bundle is trivial iff it has a section, so a Cartan principal bundle is locally trivial iff it has a local section. Thus Cartan principal bundles generalize the question of existence of local sections of subgroups.

Palais also generalizes Cartan principal bundles further by defining the notions of Cartan G-space and proper G-space, which do away with the action being free. He then proceeds to use these notions to give new proofs to several interesting theorems. See also this question on MO: http://mathoverflow.net/questions/57015/which-principlal-bundles-are-locally-trivial

What Husemoller calls a fiber bundle is the bundle associated to the Cartan principal bundle. I haven't read Husemoller that far, but I believe that such a bundle is not necessarily locally trivial.

-