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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Divisible $ H$-spaces


Author: Robert F. Brown
Journal: Proc. Amer. Math. Soc. 26 (1970), 185-189
MSC: Primary 55.40
MathSciNet review: 0261594
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Abstract: Let $ X$ be an $ H$-space with multiplication $ m$. Define, for $ x \in X,{m_2}(x) = m(x,x)$ and $ {m_k}(x) = m(x,{m_{k - 1}}(x))$, for all $ k > 2$. If $ {m_k}(x) = y$, then $ x$ is called a $ k$th root of $ y$. The $ H$-space $ (X,m)$ is divisible if every $ y$ in $ X$ has a $ k$th root for each $ k \geqq 2$. We prove that if $ X$ is a compact connected topological manifold without boundary, then $ (X,m)$ is divisible and, in fact, that every $ y$ in $ X$ has at least $ {k^\beta }k$th roots for each $ k \geqq 2$, where $ \beta $ is the first Betti number of $ X$.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9939-1970-0261594-3
Keywords: $ k$th root, compact topological manifold, Hopf algebra, exterior algebra
Article copyright: © Copyright 1970 American Mathematical Society