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I will talk about a generalization of a classical notion of an H-space. Namely, a path-connected Hausdorff topological space X is called an nH-space, n > 1, if it admits an n-valued multiplication with unit axiom. This means that there exist a continuous map μ: X ×X → Sym^nX = X^n/S_n, and a point e ∈ X such that μ(x,e) = μ(e,x) = [x,x,...,x] for all x ∈ X. It can be proven that any simply-connected finite CW-complex X of dimension d admits a structure of an nH-space for all n ≥ d. I will present the following two results: (A) a suspension over an arbitrary connected finite or countable polyhedron is an 2H-space; (B) any smoothable homology sphere is a 2H-space. References 1. Gugnin D.V. Any suspension and any homology sphere are 2H-spaces, Proceedings of the Steklov Institute of Mathematics, 2022, V. 318, P. 45-58.