Homotopy-everything 𝐻-spaces

Abstract

An H-space is a topological space X with basepoint e and a multiplication map m: X 2 = XXX->X such that e is a homotopy identity element, (We take all maps and homotopies in the based sense. We use k-topologies throughout in order to avoid spurious topological difficulties. This gives function spaces a canonical topology.) We call X a monoid if m is associative and e is a strict identity.In the literature there are many kinds of ü-space: homotopyassociative, homotopy-commutative, ^««-spaces [3], etc. In the last case part of the structure consists of higher coherence homotopies. In this note we introduce the concept of homotopy-everything H-space {E-space for short), in which all possible coherence conditions hold. We can also define £-maps (see §4). Our two main theorems are Theorem A, which classifies E-spaces, and Theorem C, which provides familiar examples such as BPL. Many of the results are folk theorems. Full details will appear elsewhere.A space X is called an infinite loop space if there is a sequence of spaces X n and homotopy equivalences X n c^tiX n+ i for n^O, such that X = Xo. THEOREM A. A CW-complex X admits an E-space structure with TQ(X) a group if and only if it is an infinite loop space. Every E-space X has a(i 'classifying space" BX, which is again an E-space.

Multivariable cochain operations and little 𝑛-cubes

“…Let C n be the little n-cubes operad ( [16], [4]). In this section we define a suboperad S n of S which will turn out to be quasi-isomorphic (in the category of chain operads over Z) to S * C n .…”

Operads and Cosimplicial Objects: An Introduction

“…In the ⇐= direction, the first version was proved by Stasheff [34], assuming that Y is connected and using a particular non-symmetric operad, now called the Stasheff operad (but the concept of operad hadn't yet been defined at that time). Boardman and Vogt proved the ⇐= direction for general A ∞ operads (except that they used PROP's instead of operads), but still assuming Y connected, in [5,6]. May defined the concept of operad in [24] and proved the ⇐= direction for connected Y ; he proved the general version (for group-complete actions) in [25].…”

“…An E ∞ operad is an operad O for which each space O(k) is weakly equivalent to a point. 5 A space with an action of an E ∞ operad should be thought of as "commutative up to all higher homotopies. "…”

“…For example, if n = 2 and c ∈ C 2 (2), d 1 ∈ C 2 (3) and d 2 ∈ C 2 (2) are represented by The definition of C n and its composition maps is due to Boardman and Vogt [5,6]. They were working in a somewhat different context (PROP's instead of operads).…”

A Note on Infinite Loop Space Multiplications

“…In [1] and [2] J. M. Boardman and I proved that an //-space X is an infinite loop space iff its multiplication enjoys nice properties concerning associativity and commutativity. These properties were described in terms of universal algebra, and the necessary and sufficient condition for X to be an infinite loop space essentially boils down to the fact that the morphism spaces £(n, 1) of the PROP £ encoding the //-structure of X be contractible (for the definition of a PROP see [2, Definition 2.44]).…”