Let $G$ be any topological group. Then we can form the infinite join $E_G$ of $G$, i.e. the colimit $G*G*G\cdots$.

Is $E_G$ contractible?

I mean it is clear that $E_G$ is weakly contractible, but I dont see why it should be contractible if $G$ is not a CW-space.

  • 1
    $\begingroup$ perhaps do you mean contractible? $\endgroup$ – johndoe Dec 4 '13 at 8:41
  • $\begingroup$ If I'm not mistaken, you only need $E_G$ to be weakly contractible in order to view it as a universal $G$-bundle, $G$ arbitrary. I guess $E_G$ might indeed be non-contractible for wild $G$. $\endgroup$ – johndoe Dec 4 '13 at 9:26
  • $\begingroup$ Do you know any example? $\endgroup$ – Oliver Straser Dec 4 '13 at 9:44
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    $\begingroup$ I can't see why Andrew Stacey's argument (see mathoverflow.net/questions/198/… ) would not work ... if $G$ is a subspace of some vector space $V$ over $\mathbf R$ you can see the Milnor construction as the convex hull of the individual copies of $G$ in the product $V\times V\times ...$ and apply the shift on that ... or do I miss something ? $\endgroup$ – few_reps Dec 4 '13 at 9:49
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    $\begingroup$ @O.Straser By Proposition 14.4.6 in T. tom Dieck: Algebraic Topology the space $E_{G}$ is indeed contractible and the assumption on $G$ is only that it is a topological group (as far as I can understand). The argument is pretty much the same as the one mentioned in the above comment by few_reps. $\endgroup$ – Oldřich Spáčil Dec 4 '13 at 15:33

Oliver Straser is correct in that Milnor himself in 1956 [1] only showed his model for $EG$ is weakly contractible (that is, all its homotopy groups vanish) with his coarse topology on the join. (For any compact Hausdorff spaces $X$ and $Y$, by the tube lemma, note that Milnor's $X \circ Y$ equals the quotient-topology join $X \ast Y$.)

Historically, for any topological group $G$, the first proof in the literature of the contractibility of Milnor's $EG := G^{\circ \aleph_0}$ is contained within the two-page proof of Theorem 8.1 in Dold's 1963 article [2].

A few years later in 1966, tomDieck in Section 7 of [3] offers a more conceptual argument.

Contemporarily, I agree that the comment of Oldřich Spáčil is an acceptable answer, based on the simplified proof that tomDieck later gives in Proposition 14.4.6 of his 2008 book [4], which rehashes his earlier works.

[1] : John Milnor, Construction of universal bundles II, Annals Math 63(3):430–436, 1956.

[2] : Albrecht Dold, Partitions of unity in the theory of fibrations, Annals Math 78(2):223–225, 1963.

[3] : Tammo tomDieck, Klassifikation numerierbarer Bündel, Arch Math (Basel) 17:395–399, 1966.

[4] : Tammo tomDieck, Algebraic Topology, 2008.


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