Higher homotopy commutativity and the resultohedra

Abstract

We define a higher homotopy commutativity for the multiplication of a topological monoid. To give the definition, we use the resultohedra constructed by Gelfand, Kapranov and Zelevinsky. Using the higher homotopy commutativity, we have necessary and sufficient conditions for the classifying space of a topological monoid to have a special structure considered by Félix, Tanré and Aguadé. It is also shown that our higher homotopy commutativity is rationally equivalent to the one of Williams.

Splitting of gauge groups

“…Then we expect that the loop spaces of H(k, l)-spaces form a new class of higher homotopy commutativity. Kawamoto and Hemmi [12] introduced H k (n)-spaces in order to unify Aguadé's T k -spaces [1] and Félix and Tanré's H(n)-spaces [6]. They also introduced higher homotopy commutativity called C k (n)-spaces in order to describe H k (n)-spaces by higher homotopy.…”

“…Later, Williams [26] introduced another kind of higher homotopy commutativity using associahedra in section 2. Recently, Hemmi and Kawamoto [12] studied a relation between higher homotopy commutativity, Aguadé's T k -spaces [1] and Félix and Tanré's H(n)-spaces [6]. In order to relate them, they introduced H k (n)-spaces and C k (n)-spaces.…”

“…In order to relate them, they introduced H k (n)-spaces and C k (n)-spaces. H k (n)-spaces collect Aguadé's T k -spaces and Félix and Tanré's H(n)-spaces whose definition is given by a sequence of H(k, l)-spaces for k + l = n (see [12]). On the other hand, C k (n)-spaces are defined as follows by using Gel'fand, Kapranov and Zelevinsky's polytopes called resultohedra (see [9], [10] for the definition of resultohedra).…”

“…The resultohedron N m,n is an (m+n−1)-dimensional polytope in R Gel'fand, Kapranov and Zelevinsky [10] described the face maps On the other hand, Hemmi and Kawamoto [12] described the degeneracy maps …”

“…Now a C k (n)-space is defined by a coherent sequence of maps Q r,s : N r,s × X r+s → X for a topological monoid X, r + s ≤ n and s ≤ k (see [12] for a precise definition). The main result of [12] is: As noted above, the definition of an H k (n)-space is a collection of H(k, l)-spaces for k + l ≤ n and, actually, the proof of Theorem 7.1 is done by collecting constructions on H(k, l)-spaces.Then, by defining C(k, l)-spaces as follows, which is a modification of that of C k (n)-spaces, we obtain Theorem 1.6.…”