# 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.

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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