Prof. Alexandru Suciu MATH 7321 TOPOLOGY 3 Spring 2017 A note on cofibrations

1. Cofibrations In this short note, we go over some classical properties of cofibrations, and the way they relate to wedges, products, and H-spaces. We start with an alternate definition of cofibration, due to Strøm [1] (see also Strom [2]).

Definition 1.1. A subspace A ⊆ X is a strong neighborhood deformation retract if there is an open set U ⊆ X containing A, and a strong deformation retraction r : U → A.

This condition (closely related to the notion of ‘good pair’ we encountered before) is almost enough to guarantee that the inclusion of A into X is a cofibration.

Theorem 1.2 ([1]). Let A ⊆ X be a closed subspace. The inclusion i: A,→ X is a cofibration if and only if there is a map φ: X → I such that A = φ−1(0) is a strong deformation retract of U := φ−1([0, 1)).

Remark 1.3. As noted previously, all cofibrations are injective. Moreover, suppose that X is compactly generated, or, more generally, X is weak Hausdorff (that is, if f : K → X is a map from a compact Hausdorff space K, then f(K) is closed in X). In that case, we know that the following holds: if i: A,→ X is a cofibration, then i(A) is closed in X. Hence, under one of those mild assumptions on X, the above theorem gives a complete characterization of cofibrations.

The next result shows that the natural map that arises when we form the product of a pair of cofibrations is again a cofibration.

Proposition 1.4. If the maps f : A → X and g : B → Y are cofibrations, then the map

f × idY ∪ idX ×g :(A × Y ) ∪ (X × B) → X × Y is also a cofibration.

A proof (which relies heavily on Theorem 1.2) can be found in [2, Proposition 5.24]. As noted there, this result should not be viewed as a formal consequence of the notion of cofibration in a model category, but rather, as a property specific to the topological category. MATH 7321 Handout 1 Spring 2017

2. Well-pointed spaces

Definition 2.1. A pointed space (X, x0) is said to be well-pointed if its basepoint is non-degenerate, i.e., the inclusion {x0} ,→ X is a cofibration.

Example 2.2. Let X = {0} ∪ {1/n | n ∈ N}. Then 0 is degenerate basepoint for X.

Corollary 2.3. Let (X, x0) and (Y, y0) be two well-pointed spaces. Then the inclu- sion X ∨ Y,→ X × Y is a cofibration.

Proof. Set X ∨ Y = ({x0} × Y }) ∪ (X × {y0}), and apply Proposition 1.4 to the cofibrations {x0} → X and {y0} → Y . 

3. H-spaces Recall that an H-space is a pointed space (X, e) endowed with a ‘multiplication’ map µ: X × X → X such that e is a homotopy unit, that is, the maps λ, ρ: X → X given by λ(x) = µ(e, x) and ρ(x) = µ(x, e) are both homotopic to idX . Equivalently, if ∇: X ∨ X → X is the ‘fold’ map (restricting to idX on each factor of the wedge), then the diagram X ∨ X ∇ (1) i  µ X × X /' X commutes up to homotopy. Proposition 3.1. Let (X, e) be a well-pointed H-space. Then the multiplication map µ: X × X → X is based-homotopic to another multiplication map, µ0 : X × X → X, which has e as a strict unit. Proof. Since X is a H-space, there is a based homotopy F :(X ∨ X) × I → X from µ ◦ i to ∇. On the other hand, since (X, e) is well-pointed, Corollary 2.3 insures that the inclusion i: X ∨X,→ X ×X is a cofibration. Consider now the commuting diagram

i X ∨ X / X × X µ

x (2) j0 X j0 9 f F G

 i×id  (X ∨ X) × I / (X × X) × I MATH 7321 Handout 1 Spring 2017 where j0(y) = (y, 0). By the homotopy extension property for the cofibration i, the homotopy F lifts to a homotopy G, as indicated above. Let µ0 : X × X → X be the map given by µ(y) = G(y, 1). By the (strict!) commutativity of the diagram, µ0(x, e) = ∇(x, e) = x, 0 and similarly, µ (e, x) = x. This completes the proof.  To recap: if we work in the category of weak Hausdorff, well-pointed spaces, then the inclusion of the wedge into the product is always a cofibration, and H-spaces always can be assumed to have a strict unit.

References [1] Arne Strøm, Note on cofibrations, Math. Scand. 19 (1966), 11–14. MR0211403 [2] Jeffrey Strom, Modern classical homotopy theory, Grad. Stud. Math., vol. 127, Amer. Math. Soc., Providence, RI, 2011. MR2839990