Blakers–Massey theorem

Remarks

Here a pair $(Y, Z)$ is $k$-connected if every path component of $Y$ meets $Z$ and $\pi_i(Y, Z, z) = 0$ for $1 \le i \le k$ and all $z \in Z$, as for n-connected spaces. The theorem is also called the homotopy excision theorem. For homology, excision gives $H_i(A, C) \cong H_i(X, B)$ in all degrees; for homotopy groups excision holds only in the range above. The proof reduces to the case where $A$ and $B$ are obtained from $C$ by attaching cells. It then works cell by cell with a general position argument in the cubes $I^i$. Applied to the two cones $A$ and $B$ that form the suspension of an $(n-1)$-connected complex, it gives the Freudenthal suspension theorem.

Going deeper

For a student

The theorem is the reason homotopy theory becomes homology-like in the stable range. It implies Freudenthal suspension, and it compares a cofibre sequence with a fibre sequence in a range of degrees. A version for cubical diagrams of spaces, due to Goodwillie, is a basic tool in the calculus of functors.