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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The number of roots in a simply-connected $H$-manifold
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by Robert F. Brown and Ronald J. Stern PDF
Trans. Amer. Math. Soc. 170 (1972), 499-505 Request permission

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 $j \leqslant k$.
References
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • 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