## Commutative algebra [Lecture notes] by Jason P. Bell PDF

By Jason P. Bell

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**Example text**

22 / cofibrations and fibrations (iv) For g ∈ π1 (E, e) and y ∈ πn (E, e), p∗ (gx) = gp∗ (x). (v) For g ∈ π1 (E, e) and z ∈ πn (B, b), ∂(gz) = g∂(z). proof. We deﬁne the actions of π1 (E, e) on πn (Fb , e), πn (E, e) and πn (B, b) to be those given by λ(E,p) , λ(E,r) and λ(B,r) ◦ p∗ , respectively. We let π1 (F, e) act on these groups by pullback along ι∗ : π1 (Fe , e) −→ π1 (E, e). By (i) and inspection, this implies that its actions are given by λ(Fe ,r) , λ(E,r) ◦ ι∗ , and the trivial action, respectively.

That is, the maps y → ∗ · y and y → y · ∗ are both homotopic to the identity map. Equivalently, the composite of the inclusion Y ∨ Y −→ Y × Y and the product is homotopic to the fold map ∇ : Y ∨ Y −→ Y, which restricts to the identity map on each wedge summand Y. Using our standing assumption that basepoints are nondegenerate, we see that the given product is homotopic to a product for which the basepoint of Y is a strict unit. Therefore, we may as well assume henceforward that H-spaces have strict units.

The following diagram commutes up to the natural transformation Uλ(E,p) −→ λ(B,p) p∗ given by the inclusions Fe −→ Fb . p∗ E λ(E,p) G B λ(B,p) HoT G HoU U Here U is the functor obtained by forgetting basepoints. 5. actions of fundamental groups in fibration sequences / 21 proof. It remains to prove the last statement. By deﬁnition, for a path β : b −→ b , λ(B,p) [β] = [β˜1 ], where β˜ is a homotopy that makes the following diagram commute. ⊂ G nT E n n n n ∩ n n n G I G B Fb × I Fb × {0} β˜ π2 p β ˜ 1), β˜ If b = p(e), we may restrict Fb to its components Fe .

### Commutative algebra [Lecture notes] by Jason P. Bell

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