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Topological applications of graded Frobenius $n$-homomorphisms, II

D. V. Gugnin

2013
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Transactions of the Moscow Mathematical Society
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This paper strengthens Theorems 3.1.7 and 3.2.4 in Topological applications of graded Frobenius n-homomorphisms, by D. V, Gugnin, Tr. Mosk. Mat. Obs. 72 (2011), no. 1, 127-188; English transl., Trans. Moscow Math. Soc. 72 (2011), no. 1, 97-142. The improved version of Theorem 3.1.7 allows us to use integral techniques when working with rational cohomology algebras of nH-spaces. We introduce a rather broad class of even-dimensional manifolds M and, using integrality conditions, we show that
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... , we show that those manifolds do not admit a 2-valued multiplication with identity. In particular, we show that complex projective spaces CP m , m ≥ 2, are not 2H-spaces. This fact has only been known for CP 2 . § 1. Introduction In Chapter 3 of [4], we investigated n-valued topological groups X and, more generally, nH-spaces, nH-monoids, and nH-groups. Those spaces X were subject to the following restrictions: locally contractible, paracompact, finite-dimension of rational cohomology, and a paracompact X n 2 . One of the goals of the present paper is to extend Theorems 3.1.7 and 3.2.4 of [4] to a less exotic, from the point of view of algebraic topology, class of spaces. In fact, we consider Hausdorff spaces X which are homotopy equivalent to a CWcomplex and such that dim H q (X; Q) < ∞ ∀q ≥ 0. It turns out that the existence of a cellular structure (for some space homotopy equivalent to X) and, accordingly, cellular cohomology, allows for a substantial strengthening of Theorem 3.1.7. More precisely, we show that, for an nH-space X, the diagonal Δ : H * (X; Q) → H * (X; Q) ⊗ H * (X; Q) of the graded n-Hopf algebra sends integral rational cohomology classes α ∈ H * (X; Q) to integral elements of the tensor product H * (X; Q) ⊗ H * (X; Q). The obtained integrality condition allows us to show that manifolds in a rather wide class M, introduced in this paper, do not admit a 2-valued multiplication with identity. In particular, we show that projective spaces CP m , m ≥ 2, do not admit a 2-valued multiplication with identity. Previously, this fact had been known only for CP 2 (see [2, p. 340]), even though it was announced for all m ≥ 2. Recall, however, that CP m , m ≥ 1, carries a structure of (m + 1)!-valued topological groups (see [1, p. 64 ]). § 2. Main integrality lemma Consider an arbitrary compact polyhedron X. Let K be the field of coefficients, char K = 0 or p, p > n. In this case, one can easily compute the cohomology ring H * (Sym n X; K) of the nth symmetric power of X. The answer is given by 2010 Mathematics Subject Classification. Primary 13A02, 16T05, 55P45, 57N65.

doi:10.1090/s0077-1554-2013-00201-0
fatcat:z7qgmqhmabhixhciab4sm3hviq