Talk:Aspherical space

Eilenberg-MacLane

Such spaces are also called K(G, 1) spaces, or Eilenberg-MacLane spaces.

Such spaces are said to be the classifying space for the group G. Milnot gives a construction for such a space in his book Characteristic Classes.

I think that the remark about symplectically/symplectic aspherical was useful - I encountered both variants, and so clarifying what it means doesn't hurt.

Also, in examples, I don't get how non-orientable case follows from orientable case.Sirix 09:51, 20 April 2007 (UTC)

Here is the argument, in more detail. Every non-orientable manifold M admits an orientation double cover, M^*. For example, RP² has orientation double cover S², the Klein bottle has orientation double cover the torus, etc. If M is a non-orientable surface other than RP², then its orientation double cover, M^*, will be an orientable surface of genus at least 1 (the precise genus can be worked out by an Euler characteristic argument); hence, M^* is aspherical; hence M is aspherical. More generally, a non-orientable manifold M is aspherical if and only if its double cover, M^*, is aspherical. At any rate, I could put this in more carefully in the article, if you think it would be useful. I can also probably find a reference where this is mentioned, though I think the argument is so clean and self-contained that it could also be written out (briefly) here. Turgidson 10:25, 20 April 2007 (UTC)

Symplectic

The expression

> Some references[2] drop the requirement on c1 in their definition of "symplectically aspherical."

should be changed to a more precise one. Sirix (talk) 14:39, 1 September 2010 (UTC)

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Introduction is not well written

In the three paragraphs of the introductory section, we are told a) that an aspherical space is defined as one whose higher homotopy groups are trivial; b) that if "one works with CW-complexes" an aspherical space is one whose universal cover is contractible; and c) that an aspherical space is "by definition", a K(π, 1) space.

We are left wondering whether these three concepts are the same or not.

If an aspherical space is synonymous with the space being an Eilenberg-Mac Lane space, then the article should come right out and say so.

If a CW-complex is aspherical if and only if its universal cover is contractible, again, the article should come right out and say so.

Otherwise readers are left to wonder why the article did not come right out and answer these obvious questions.50.205.142.35 (talk) 00:16, 3 January 2020 (UTC)

Shouldn't the statement be stronger?

The introductory section states the following:

"Each aspherical space X is, by definition, an Eilenberg–MacLane space of type $K(G,1)$, where $G=\pi _{1}(X)$ is the fundamental group of X."

Of course this is true. But isn't it in fact the case that for any topological space X, the property of being aspherical is equivalent to being a K(π₁(X), 1) ???

In this case, it is not good enough for the article to state merely that "Each aspherical space is, by definition" a K(π₁(X), 1), because this way of wording the statement omits the fact that the two concepts are in fact equivalent.47.44.96.195 (talk) 17:38, 20 October 2020 (UTC)