# Simple space

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In [algebraic topology](/source/algebraic_topology), a branch of [mathematics](/source/mathematics), a '''simple space''' is a [connected](/source/connected_space) [topological space](/source/topological_space) that has a homotopy type of a [CW complex](/source/CW_complex) and whose [fundamental group](/source/fundamental_group) is [abelian](/source/abelian_group) and [acts](/source/group_action) trivially on the homotopy and homology of the universal covering space,<!-- what about principal fibration? --> though not all authors include the assumption on the homotopy type.

== Examples ==

=== Topological groups ===
For example, any [topological group](/source/topological_group) is a simple space (provided it satisfies the condition on the homotopy type).

=== Eilenberg-Maclane spaces ===
Most [Eilenberg-Maclane space](/source/Eilenberg-Maclane_space)s <math>K(A,n)</math> are simple since the only nontrivial homotopy group is in degree <math>n</math>. This means the only non-simple spaces are <math>K(G,1)</math> for <math>G</math> [nonabelian](/source/nonabelian_group).

=== Universal covers ===
Every connected topological space <math>X</math> has an associated (universal) <!-- Expand on "universal". Is it meant as in "the terminal object in the category of simple spaces with a map to X"?  --> simple space from the universal cover <math>\pi:U_X \to X</math>; indeed, <math>\pi_1(U_X) = *</math> and the universal cover is its own universal cover.

== References ==
*[Dennis Sullivan](/source/Dennis_Sullivan), ''[http://www.maths.ed.ac.uk/~aar/books/gtop.pdf Geometric Topology]''

Category:Algebraic topology

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Adapted from the Wikipedia article [Simple space](https://en.wikipedia.org/wiki/Simple_space) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Simple_space?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.