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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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Keywords: <IMG WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="$H$">-space, <IMG WIDTH="17" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$k$">th root, coincidence-preserving homotopy
Article copyright: © Copyright 1972 American Mathematical Society