(L)-semigroup sums (Q2306120)

Summary: An (L)-semigroup \(S\) is a compact \(n\)-manifold with connected boundary \(B\) together with a monoid structure on \(S\) such that \(B\) is a subsemigroup of \(S\). The sum \(S + T\) of two (L)-semigroups \(S\) and \(T\) having boundary \(B\) is the quotient space obtained from the union of \(S \times \{0 \}\) and \(T \times \{1 \}\) by identifying the point \((x, 0)\) in \(S \times \{0 \}\) with \((x, 1)\) in \(T \times \{1 \}\) for each \(x\) in \(B\). It is shown that no (L)-semigroup sum of dimension less than or equal to five admits an H-space structure, nor does any (L)-semigroup sum obtained from (L)-semigroups having an abelian boundary. In particular, such sums cannot be a retract of a topological group.