Homotopy-everything 𝐻-spaces
1968; American Mathematical Society; Volume: 74; Issue: 6 Linguagem: Inglês
10.1090/s0002-9904-1968-12070-1
ISSN1088-9485
Autores Tópico(s)Algebraic structures and combinatorial models
ResumoAn 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.
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