(5.38) Let \(X^1\) be the 1-skeleton of \(X\). This is obtained by attaching 1-cells to a point, so \(X^1\) is just a wedge of 1-cells \(X^1=\bigvee_{1\mbox{-cells}}S^1\). By induction, using the computation from last time, \(\pi_1(X^1)=\bigstar_{1\mbox{-cells}}\mathbf{Z}\) (i.e. the free product of copies of the integers, one for each 1-cell, in other words, the group with one generator for each 1-cell and no relations).

(7.42) What happens when we attach a 2-cell?

Claim: If \(Y\) is any space and \(Z\) is obtained from \(Y\) by attaching a 2-cell \(e\) with some attaching map \(\varphi\) then \(\pi_1(Z)=\pi_1(Y)/N(\partial e)\), where \(N(\partial e)\) is the normal subgroup generated by the boundary of \(e\) (that is \([\varphi]\in\pi_1(Y)\)). This is equivalent to imposing the relation \(\partial e=1\).

(9.48) Proof of claim: This follows from Van Kampen's theorem. If we decompose \(Z=U\cup V\) where \(U\) is the interior of the 2-cell \(int(e)\) and \(V\) is the union \(Y\cup (e\setminus\{0\})\) (that is, \(V\) is everything except one point in the interior of \(e\)).

• The intersection \(U\cap V\) is \(e\setminus\{0\}\), the punctured 2-cell. That is homotopy-equivalent to a circle.
• \(U\) is contractible.
• \(V\) is homotopy equivalent to \(Y\) (retracting the punctured 2-cell down onto its boundary in \(Y\)).

(11.45) Van Kampen's theorem tells us that \(\pi_1(Z)=\pi_1(Y)\star_{\mathbf{Z}}\{1\}\). In other words, we take \(\pi_1(Y)\) and we add an amalgamated relation \(\partial e=1\) (\(\partial e\) generates \(\pi_1(U\cap V)\): in \(\pi_1(U)\) is becomes trivial and in \(\pi_1(V)\) it becomes \(\partial e\)).

(13.28) What about adding higher dimensional cells? By the same argument, where \(e\) is an \(n\)-cell with \(n>2\), we get \[\pi_1(Z)=\pi_1(Y)\star_{\{1\}}\{1\},\] because \(U\cap V\) is now homotopy equivalent to the \((n-1)\)-sphere, which is simply-connected (so there is no new amalgamated relation).