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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The number of roots in a simply-connected $ H$-manifold


Authors: Robert F. Brown and Ronald J. Stern
Journal: Trans. Amer. Math. Soc. 170 (1972), 499-505
MSC: Primary 55D45; Secondary 57A15, 57C15
DOI: https://doi.org/10.1090/S0002-9947-1972-0307227-5
MathSciNet review: 0307227
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Abstract: An $ H$-manifold is a triple $ (M,m,e)$ where $ M$ is a compact connected triangulable manifold without boundary, $ e \in M$, and $ m:M \times M \to M$ is a map such that $ m(x,e) = m(e,x) = x$ for all $ x \in M$. Define $ {m_1}:M \to M$ to be the identity map and, for $ k \geqslant 2$, define $ {m_k}:M \to M$ by $ {m_k}(x) = m(x,{m_{k - 1}}(x))$. It is proven that if $ (M,m,e)$ is an $ H$-manifold, then $ M$ is simply-connected if and only if given $ k \geqslant 1$ there exists a multiplication $ m'$ on $ M$ homotopic to $ m$ such that $ {m'_j}(x) = e$ implies $ x = e$ for all $ j \leqslant k$.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0307227-5
Keywords: $ H$-space, $ k$th root, coincidence-preserving homotopy
Article copyright: © Copyright 1972 American Mathematical Society