Any Suspension and Any Homology Sphere Are $$2H$$-Spaces
Dmitry Vladimirovich Gugnin · Proceedings of the Steklov Institute of Mathematics · 2022
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 $$X=S^{2k+1}$$ , $$k e 0,1,3$$ , 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 $$\mathrm{Sym}^n f\colon\, \mathrm{Sym}^n X \to \mathrm{Sym}^n\kern1pt Y$$ also induces an isomorphism in integral homology.