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This is a qual question and I have no idea how to begin. Help is much appreciated.

Let $X$ and $Y$ be path connected spaces, and let $f : X \rightarrow Y$ be a continuous map. The mapping cone $Cf$ of $f$ is defined to be the quotient space of $Y \coprod (X \times [0, 1])$ obtained by identifying each point $(x,0), x \in X,$ with the point $f(x) \in Y,$ and identifying all points $(x,1)$ to a single point . If $\pi_1(Cf)$ is trivial, is it necessarily true that the map on $\pi_1$ induced by f is onto? Prove or give a counterexample.

kingkongdonutguy
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1 Answers1

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I'll add some more detail to the above hint: $Cf$ can be formed by gluing the cone $CX=C(\mbox{Id}_X)$ to the 'mapping cylinder' $M_f$ which is equal to the quotient of $Y\sqcup (X\times[0,1])$ by identifying $(x,0)$ for $x\in X$ with $f(x)\in Y$. Note that $C X$ is contractible, and $M_f$ deformation retracts onto the natural subspace $Y$.

We glue these two space together on their respective embedded copies of $X$ in the natural way and so their intersection as subspace of $C_f$ is $X$. If we take a small open 'thickening' of these subspace of $Cf$ we can apply Van Kampen's theorem because the necessary spaces are path-connected and open. I'll leave the conclusion to you.

Dan Rust
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