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This is a very elementary question but I can't quite seem to track down a worthwhile source, so I was hoping someone more knowledgeable than I could lend their superiority.

In Moore & Schochet's book Global Analysis on Foliated Spaces, the following "notational definition" is given:

Let $F$ be a space, let $B^p$ be a manifold..., and let $\tilde B \to B$ denote the universal cover. Suppose given a homomorphism $\varphi:\pi_1(B)\to\operatorname{Homeo}(F)$. Form the space $$M=\tilde{B}\times_{\pi_1(B)}F$$ as a quotient of $\tilde{B}\times F$ by the action of $\pi_1(B)$ determined by deck transformations on $\tilde{B}$ and by $\varphi$ on $F$.

So my question is about $\times_{\pi_1(B)}$: Surely this is an example of a more general notation, and based on the excerpt above, it seems like it's used to denote the usual direct product of two spaces modulo an action by some group?

My question, I guess: To what extent is that accurate? What are the properties of the two spaces whose product is being taken? What is the product of the action by which we're quotienting?

I've seen similar notation (in, e.g., Lawson's Quantitative Theory of Foliations) representing the Moebius strip in the form $\mathbb{R}\times_{\mathbb{Z}}\mathbb{R}$, and I've consulted a number of sources - Hatcher's Algebraic Topology, Steenrod's Topology of Fiber Bundles, etc. - and haven't quite tracked down what I'm looking for. I know there's a similar notation ($\rtimes_\varphi$) used for semidirect products in Dummitt and Foote, too...there's obviously something bigger going on that I don't know enough about to know about.

Any information (including an indication of relevant literature) would be greatly appreciated!

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  • $\begingroup$ Did you ever find an answer? I was looking for the same and came up with your question. $\endgroup$ – X1921 Dec 8 '18 at 11:43
  • $\begingroup$ @X1921 - I've included an answer below. The details may be a bit rough, but I think the ideas are all there. Hope it helps! $\endgroup$ – cstover Dec 8 '18 at 13:02
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Because someone asked for a follow-up, I'll state what I (think I) know.

If $\tilde{B}\to B$ is the universal cover, then the fundamental group $\pi_1(B)$ of $B$ satisfies $\tilde{B}/\pi_1(B)\cong B$. In particular, $\pi_1(B)$ acts freely on $\tilde{B}$ by deck transformations, so for $x,x'\in\tilde{B}$, $x\sim x'$ if and only if $x$ differs from $x'$ by a Deck transformation if and only if $x=g(x')$ for some $g\in\pi_1(B)$.

Now, if $\varphi:\pi_1(B)\to\operatorname{Homeo}(F)$ is a homomorphism, then each $g\in\pi_1(B)$ also determines a homeomorphism $\varphi(g):F\to F$.

So, to form the quotient $\tilde{B}\times_{\pi_1(B)} F$, you're considering the relation $(\tilde{B}\times F)/\sim$ where $(x,y)\sim(x',y')$ if and only if $x,x'\in\tilde{B}$ satisfy $x=g(x')$ and $y,y'\in F$ satisfy $y=\varphi(g)(y')$ for some $g\in\pi_1(B)$.

To me, this is analogous to the formation of a mapping torus (see https://en.wikipedia.org/wiki/Mapping_torus).

As for this question:

Surely this is an example of a more general notation, and based on the excerpt above, it seems like it's used to denote the usual direct product of two spaces modulo an action by some group?

My question, I guess: To what extent is that accurate? What are the properties of the two spaces whose product is being taken? What is the product of the action by which we're quotienting?

I think the notations $\times_G$, $\rtimes_\varphi$, etc. almost always refer to taking some product in some category (see https://en.wikipedia.org/wiki/Product_(category_theory)) and quotienting out by some morphism relationship in that category (https://en.wikipedia.org/wiki/Quotient_category). Such has been my experience, at least.

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