By J.P. May
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Extra info for The Geometry of Iterated Loop Spaces
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There is an E∞ operad L such that L acts on F˜ (so as to induce the smash product) in such a manner that the composition product F˜ × F˜ → F˜ is a morphism of L-spaces. Of course, there is a distributive law relating the loop product φ to the composition product, namely φ(f1 , f2 ) ◦ S g = φ(f1 ◦ S g, f2 ◦ S g) for f1 , f2 ∈ Ωn X and g ∈ F˜ (n − 1). 8. 2, simply by writing down the definitions. 9. For X ∈ T and all positive integers m, n, and j, the following diagram is commutative: Cm (j) × (Ωm+n X)j × F˜ (n) 1×1×∆(S )m θm,n ×(S )m cm+n Cm (j) × (Ωm+n X)j × F˜ (m + n)j 1×u Cm (j) × (Ωm+n X × F˜ (m + n))j / Ωm+n X × F˜ (m + n) Ωm+n O X θm,j 1×cjm+n / C (j) × (Ωm+n X)j m We can pull back the composition product along the approximation maps αn , but this fact is slightly less obvious.
G starts at the identity and ends in Fj−1 En (X, A) since vi (c) ≤ 0 for at least one i in each c. Note, however, that G cannot be extended over all of Fj En (X, A). Now assume that there exists > 0 such that g(I × u−1 [0, ]) ⊂ u−1 [0, 1). If x is compact, then there exists such an by an easy exercize in point-set topology; if X = T A, (j, v) represents (A, ∗) as an NDR-pair, and u[a, s] = v(a) · s, h(t, [a, s]) = [j(t, a), s], and g(t, [a, s]) = [a, s − st], then any < 1 suffices. Define a homotopy H : I × Fj En (X, A) → Fj En (X, A) by H(t, z) = z for z ∈ Fj−1 En (X, A) and by G(t, [c, y]) if uj (y) ≥ /2 H(t, [c, y]) = 2t·uj (u) , [c, y] if uj (y) ≤ /2 G 7.
Gq gq+1 , gq+1 ) and −1 αq−1 (g1 , . . , gq+1 ) = [g1 g2−1 , g2 g3−1 , . . , gq gq+1 ]gq+1 . Visibly these are inverse functions. For g ∈ G, we have ([g1 , . . , gq ]gq+1 )g = [g1 , . . , gq ]gq+1 g and (g1 , . . , gq+1 )g = (g1 g, . . , gq+1 g), and α and α−1 are thus visibly G-equivariant; they commute with the face and degeneracy operators by similar inspections. 9 and the previous result we have the following observation. 4. Let X ∈ U and let η : ∗ → X be any map in U. Define hi : Dq (X) → Dq+1 (X), 0 ≤ i ≤ q, by the formula hi = si0 (η × 1q−1 )∂0i : X q → X q+1 .