# Twisting Products in Hatcher's Algebraic Topology

In Allen Hatcher's book Algebraic Topology, in several places is used the terminology 'twisted product'; eg on page 338:

Among other things, fibrations allow one to describe, in theory at least, how the homotopy type of an arbitrary CW complex is built up from its homotopy groups by an inductive procedure of forming 'twisted products' of Eilenberg–MacLane spaces.

or page 393 in the context of Postnikov towers

To the extent that fibrations can be regarded as twisted products, up to homotopy equivalence, the spaces $$X_n$$ in a Postnikov tower for $$X$$ can be thought of as twisted products of Eilenberg-MacLane spaces $$K(\pi_n X,n)$$.

What does Hatcher mean by 'twisted products' here?

• I don't have my copy of Hatcher at hand at the moment, but he's probably just referring to a fiber bundle over the Eilenberg-MacLane spaces; that is, he's taking a twisted product of $X$ and $Y$ to be a bundle of $X$ with fiber $Y$ (though unfortunately this isn't symmetric in $X$ and $Y$). It's not uncommon in referring to certain common $3$-manifolds, for example. Feb 4, 2022 at 0:03
• He means fiber bundles. Feb 4, 2022 at 0:17
• Related to math.stackexchange.com/q/2090578. Feb 4, 2022 at 0:20
• so it's a suggestive terminology arising/motivated by geometry of the Mobius strip as archetypical example where any fiber bundle can be obtained if we 'twist' enough times? Feb 4, 2022 at 0:23
• There is a result that the category of fiber bundles and fiberwise maps up to fiberwise homotopy is equivalent to the category of fibrations and fiberwise maps up to homotopy. If by definition twisted products refer to fiber bundles, this result is what it means to say a fibration is a fiber bundle up to homotopy. Feb 4, 2022 at 1:35

Unfortunately Hatcher does not give an explicit definition of a "twisted product". However, here are two quotations which help to clarify it.

1. From the section "Fiber Bundles" on p. 375 :

General fiber bundles can be thought of as twisted products. Familiar examples are the Möbius band, which is a twisted annulus with line segments as fibers, and the Klein bottle, which is a twisted torus with circles as fibers.

1. From section 4.3 "Connections with Cohomology" on p. 393 :

The most geometric interpretation of the phrase "twisted product" is the notion of fiber bundle introduced in the previous section.

The total space $$E$$ of a fiber bundle $$p : E \to B$$ over a space $$B$$ with fiber $$F$$ looks locally like an ordinary product $$U \times F$$, where $$U \subset B$$ is a suitable open subset, but globally it is in general not the product $$B \times F$$. Looking locally like $$U \times F$$ means that there is a homeomorphism $$h : p^{-1}(U) \to U \times F$$ such that $$proj_U \circ h = p \mid_{p^{-1}(U)}$$.

The fibers of $$p$$ may be twisted. That is, given two local product representations $$h_U : p^{-1}(U) \to U \times F$$ and $$h_V : p^{-1}(V) \to V \times F$$, then on $$U \cap V$$ we get a fiber-preserving homeomorphism $$\phi = h_V^{-1} \circ h_U : (U \cap V) \times F \to (U \cap V) \times F$$ which is not necessarily the identity. This means that $$\phi$$ twists the fibers.