Any Suspension and Any Homology Sphere Are $2H$-Spaces

We prove that the reduced suspension $X = \Sigma Y$ over any finite or countable connected polyhedron $Y$ can be endowed with a two-valued multiplication $\mu \colon X\times X \to \mathrm {Sym}^2 X$ satisfying the unit axiom: $\mu (e,x) = \mu (x,e) = [x,x]$ for all $x\in X$. If $X$ is a sphere $S^m$, $m = 1,3,7$, this is a classical result; for $X=S^2$, this is V. M. Buchstaber's theorem of 1990; and for $X=S^{2k+1}$, $k\ne 0,1,3$, this is our theorem of 2019. We also prove a similar statement for all $X$ that are smoothable homology spheres of arbitrary dimension and for $X=\mathbb R\mathrm P^m$, $m\ge 2$. The proof of one of the main results uses the following statement, which is of independent interest. Let $X$ and $Y$ be connected finite CW complexes and $f\colon X\to Y$ a continuous map inducing an isomorphism in integral homology. Then, for any $n\ge 2$, the map $\mathrm {Sym}^n f\colon \mathrm {Sym}^n X \to \mathrm {Sym}^n\kern 1pt Y$ also induces an isomorphism in integral homology.