In algebraic topology, a branch of mathematics, a '''simple space''' is a connected topological space that has a homotopy type of a CW complex and whose fundamental group is abelian and acts 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 is a simple space (provided it satisfies the condition on the homotopy type).
=== Eilenberg-Maclane spaces === Most Eilenberg-Maclane spaces <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.
=== 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, ''[http://www.maths.ed.ac.uk/~aar/books/gtop.pdf Geometric Topology]''
Category:Algebraic topology
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