Talk:Aspherical space

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Mathematics rating:	Stub Class	Low Priority	Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 20:29, 21 May 2007 (UTC)

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)