TITLE: For which topological spaces $X$ can one write $X \approx Y \times Y$? Is $Y$ unique?
QUESTION [15 upvotes]: Under what conditions on topology of a given space $X$ it can be written as product space $Y\times Y$ for some other topological space $Y$. Lets call such space $Y$ a square root of X. Then given that square root of $X$ exists is it unique (within topological isomorphism) ?

REPLY [4 votes]: Here is a necessary condition for a space to have a square root.  By the Künneth theorem, if $X=Y\times Y$, then with respect to a field $F$, we must have $H_k(X;F)=\bigoplus_{i+j=k}H_i(Y;F)\otimes H_j(Y;F)$.  Therefore, associating to the space $X$ the sequence of betti numbers $b_i(X;F)=\dim_F H_i(X;F)$, we must have that the sequence $(b_i)$, if consisting of finite numbers, must be the convolution square of another sequence of non-negative integers.  
If there are only finitely many non-zero betti numbers for $X$, then the problem of existence of a convolution square root is the same as determining whether a polynomial has a polynomial square root with non-negative integer coefficients.  If there are infinitely many non-zero betti numbers, the problem becomes one related to square roots of power series, and I'm not sure if there is an easy way to tackle that.
In any event, this gives the nice result that an odd dimensional compact manifold cannot have a square root.