# Topological Meaning of semi-direct product

I know that the amalgamated free product of two groups $G\star_K H$ has a certain topological meaning. What about a semi-direct product $H \rtimes G$ ?

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Perhaps you could describe the topological meaning of the amalgamated free product you refer to first. This may help in garnering an analogous description for the semi-direct product. –  user31714 Jan 12 '13 at 20:12
@user31714 I guess that Curiosity is referring to the fact that for every two spaces with $\pi_1$ respectively $G$ and $H$ you can build up the pointed product of the two spaces whose $\pi_1$ is the free product of $G$ and $H$. –  Giorgio Mossa Jan 12 '13 at 20:49
If X and Y have homeomorphic subspace A then the fundamental group of the space obtained by gluing X and Y along A can computed using the fundamentals groups of X,Y and A using amalgamated free products. I am referring to Seifert-Van Kampen theorem and I may be not rigorous in my argument above but I hope I described the idea. –  Curiosity Jan 12 '13 at 20:55
So are you asking what topological space has fundamental group $H\rtimes G$ if $H$ and $G$ are fundamental groups of some other topological spaces? –  Alexander Gruber Jan 12 '13 at 21:16
@AlexanderGruber, Yes that is what I am asking. –  Curiosity Jan 12 '13 at 22:43

If you don't know anything about fiber bundles (or the more general family of maps fibrations) then to put it very loosely, a map $p\colon E\rightarrow B$ is a fiber bundle if for all $b\in B$, the preimage of a neighbourhood $U\ni b$ is homeomorphic to the product of some space $F$ and $U$. That is, $p^{-1}(U)\cong F\times U$. We often write that the sequence of maps $F\rightarrow E\rightarrow B$ is a fibration sequence with fiber $F$ (the first map in this sequence is just inclusion of $F$ in $E$).
The proper definition puts some restrictions on the map $p$ but the above definition is good enough to give an intuitive feel for what fiber bundles 'look like'. The important part you should take away from the definition is that the space $E$ can be seen as a kind of 'twisted product' of the spaces $F$ and $B$. You should hopefully already be able to see a loose parallel between fiber bundles and semi-direct products of groups.
A very useful property of fiber bundles of 'nice spaces' (roughly speaking, if all the spaces involved are CW-complexes then we're fine) is that the homotopy groups of the fibration sequence fit in to a long exact sequence. Now, suppose that the space $F$ is connected and $\pi_2(B)=0$. Then from this long exact sequence in homotopy, we can extract the short exact sequence $$\pi_2(B)=0\rightarrow\pi_1(F)\rightarrow\pi_1(E) \rightarrow\pi_1(B)\rightarrow 0=\pi_0(F)$$ and now it should be clear that if this short exact sequence splits, then the splitting lemma for non-ableian groups tells us that $\pi_1(E)\cong \pi_1(F)\rtimes \pi_1(B)$.
The conditions on the homotopy groups of $B$ and $F$, and the requirement that the short exact sequence splits are rather specialised. So, this is far from a universally useful construction, but hopefully it's the sort of link between fundamental groups and semi direct product of groups that you were looking for.