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Cover the attached disc by a slightly enlarged interior and a collar of its boundary, and apply the Seifert-van Kampen theorem. The collar intersection has fundamental group generated by the boundary loop, whose image in $X$ is $[\alpha ]$, while that loop becomes trivial in the disc. Van Kampen therefore quotients by the smallest normal subgroup containing $[\alpha ]$:This is the general fundamental group after attaching a 2-cell.

Start with a wedge of two circles, whose fundamental group is the free groupAttach a disc along the loop represented by $a^3{b}^{-7}$. The displayed quotient gives a connected two-dimensional CW complex $X$ with

The Seifert-van Kampen theorem says that if $X=U\cup V$, where $U,V,U\cap V$ are path connected open sets containing $x_0$, then

The cell attachment is the quotientwith the image of $x_0=f(\ast )$ as base point. For $n=2$, take one open set that deformation retracts onto $X$ together with a boundary collar, and another that consists of the interior of the disc with a collar and is contractible. Their intersection deformation retracts onto $S^1$. Its generator maps to $[f]$ in the first set and to the identity in the second. Van Kampen therefore proves the fundamental group after attaching a 2-cell formula

Applying van Kampen to the two circles givesAttach three discs along loops representing $a^2$, $b^3$, and $(ab{)}^2$. The resulting presentation complex for the symmetric group on three letters has groupSending $a$ to $(12)$ and $b$ to $(123)$ gives a surjection onto $S_3$. The relations imply $aba=b^{-1}$, so every word reduces to one ofThe presented group has at most six elements and surjects onto the six-element group $S_3$, so this map is an isomorphism.

One form of the Seifert-van Kampen theorem says that if $X=U\cup V$, with $U,V,U\cap V$ open and path-connected and containing the base point, then ${\pi }_1(X)$ is the pushout of

For the fundamental group after attaching a 2-cell, take $U$ to be $X$ together with a collar of the attached disc and $V$ to be a slightly enlarged open disc. Then $U\simeq X$, $V$ is simply connected, and $U\cap V\simeq S^1$; its generator maps to $[\alpha ]$ in ${\pi }_1(X)$. Van Kampen therefore gives

Take $X=S_a^1\vee S_b^1$, whose fundamental group is the free group $F(a,b)$, and attach one 2-cell along the loop represented by $a^2{b}^{-3}$. The resulting connected presentation complex $Y$ satisfies