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There's certainly a homotopy-theoretic analogue. A universal cover of a connected space X is (up to homotopy) a simply connected space X' and a map X' -> X which is an isomorphism on πn for n >= 2. We could next ask for a 2-connected cover X'' of X': a space X'' with πkX'' = 0 for k <= 2 and a map X'' -> X' which is an isomorphism on πn for n >= 3. The homotopy fiber of such a map will have a single nonzero homotopy group, in dimension 1--it will be a K(π2X, 1). (For the universal cover the fiber was the discrete space π1X = K(π1X, 0).)

An example is the Hopf fibration K(Z, 1) = S1 -> S3 -> S2.

Geometrically it's harder to see what's going on with the 2-connected cover than with the universal cover, because fibrations with fiber of the form K(G, 1) are harder to describe than fibrations with discrete fibers (covering spaces).

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Thanks! I hadn't thought of it in that way before, but it makes perfect sense. – j.c. Oct 17 '09 at 23:51



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There is not a universal cover for every space. One needs conditions, such as being semi-locally simply connected. – user332 Oct 18 '09 at 0:21
I think Eric is probably using the word "space" the way I use it, namely, a cofibrant-fibrant object of whatever model category you like which is Quillen equivalent to spaces. – Reid Barton Oct 21 '09 at 0:32
Or the way most people use it, namely "a manifold or CW complex or some similarly nice thing". =) – Tom Church Oct 21 '09 at 2:16


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However, the analogy to the n=1 case is not complete: While \pi_ 1(X) classifies the automorphisms over X of the universal covering space, the \pi_ n(X) don't classify the automorphisms over X of these n-connected analogues. In fact, the higher \pi_ n(X) don't seem to classify anything else than homotopy classes of maps S^n-->X.

This is one of the motivations for using n-groupoids as invariants of spaces, see the discussion here, right before the references:


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pi_n(X) will classify something when X is (n-1)-connected, though (after all there's no other information present in the n-groupoid). – Reid Barton Oct 21 '09 at 1:25

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