Π∞ preserves pointed homotopical constructions

Proposition 2.4.12. The functor \(\Pi _{\infty }\colon \Top _* \to \An _*\) has the following properties:

(1)

Given a pointed map \(f\colon X\to Y\) between spaces having the homotopy type of cell complexes, the unreduced homotopy cofiber \(C(f)\), pointed by its cone point, satisfies \(\Pi _{\infty }(C(f))\cong \cofib (\Pi _{\infty }(f))\).

(2)

If \(X\) has the homotopy type of a cell complex, the unreduced suspension \(SX\), pointed by either cone point, satisfies \(\Pi _{\infty }(SX)\cong \Sigma \Pi _{\infty }(X)\).

(3)

For a collection \((X_i)_{i\in I}\) of pointed spaces having the homotopy type of cell complexes, there is an isomorphism \(\Pi _{\infty }(\bigvee ^h_{i\in I}X_i)\cong \bigvee _{i\in I}\Pi _{\infty }(X_i)\), where the homotopy wedge \(\bigvee ^h_iX_i\) is the homotopy pushout of the span \(\bigsqcup _iX_i\leftarrow \bigsqcup _i*\to *\).

(4)

Given a pointed map \(f\colon X\to Y\) with homotopy fiber \(F_f\) over the basepoint, there is an isomorphism \(\Pi _{\infty }(F_f)\cong \fib (\Pi _{\infty }(f))\).

(5)

The loop space of a pointed space \(X\) satisfies \(\Pi _{\infty }(\Omega X)\cong \Omega \Pi _{\infty }(X)\).