Examples for a contractible pair that induce a non-surjective fundamental groups homomorphism.

Question

In Hatcher's Algebraic Topology, Proposition 1.17 says that

If a connected $X$ retracts onto a connected subspace $A$, the inclusion $i:A\hookrightarrow X$ induce a injective map $i_*:\pi_1(A)\to\pi_1(X)$. If $X$ can deformation retract to $A$, then $i_*$ is a isomorphism.

If the $X$ can retract but not deformation retract onto $A$, is the surjective of $i_*$ still holds?

The zigzag space, example in Exercise 0.6, has trivial fundamental group so it doesn't give the counterexample.

What about the weak contractible property?

Answer 1

Any product $A \times B$ will give a retract down to either factor once you pick a basepoint in the other. If both factors have interesting fundamental groups then the induced map will be injective but not surjective as it won't be iso.

Additional details...

Take $A$ and $B$ to be connected spaces and fix a point $b \in B$. Define a map $f: A \times B \rightarrow A \times \{b\}$ by $f= \mathrm{id}_A \times b$, so $f(x,y) = (x, b)$. This is clearly continuous and it clearly fixes the subspace $A \times \{b\}$ pointwise. Thus $f$ is a retract. For most examples the induced-inclusion $i_*: \pi_1(A) \to \pi_1(A\times B)$ probably isn't onto. For example, take $A=\mathbb{R}$ and $B=S^1$.