Let \(S_k\) be the space obtained by removing \(k\) disjoint open disks from the sphere \(S^2\). Form \(X_k\) by gluing \(k\) Möbius bands onto \(S_k\) , one for each circle boundary component of \(S_k\) (by identifying the boundary circle of a Möbius band homeomorphically with a given boundary component circle).
Use van Kampen’s theorem to calculate \(\pi_1 (X_k)\) for each \(k > 0\) and identify \(X_k\) in terms of the classification of surfaces.
Solution
Fix \(k\ge1\) and a basepoint in \(S_k\). For each boundary component choose an arc from the basepoint to that component, and let \(c_i\) denote the resulting based boundary loop.
1. The punctured sphere has presentation\[ \pi_1(S_k) \cong \left\langle c_1,\dots,c_k \mathrel{\Big|} c_1c_2\cdots c_k=1 \right\rangle. \]In particular, \(\pi_1(S_k)\) is free of rank \(k-1\).
Cut \(S_k\) along the chosen arcs joining the basepoint to the boundary circles. The result is a disk, whose oriented boundary reads \[ c_1c_2\cdots c_k \] after the paired arc segments cancel. Equivalently, a standard van Kampen decomposition gives one generator for each boundary loop and the single relation that their product is trivial. Eliminating \(c_k\) leaves a free group on \(c_1,\dots,c_{k-1}\).
2. Let \(M_i\) be the Möbius band glued to the \(i\)th boundary component, and let \(x_i\) be its core loop.Then\[ \pi_1(M_i)=\langle x_i\rangle\cong\ZZ, \]and the boundary circle of \(M_i\) represents \(x_i^2\), up to replacing \(x_i\) by \(x_i^{-1}\).
A Möbius band deformation retracts onto its core circle, so its fundamental group is infinite cyclic. In the usual rectangle model \[ M=[0,1]\times[-1,1]/(0,t)\sim(1,-t), \] one circuit around the boundary passes twice around the core before closing. Thus the homomorphism induced by the boundary inclusion is multiplication by \(2\) on \(\pi_1\cong\ZZ\).
The attaching homeomorphism of the boundary circle may reverse the chosen orientation, changing \(x_i^2\) to \(x_i^{-2}\). Replacing \(x_i\) by its inverse removes this sign, so we may write the attaching relation as \[ c_i=x_i^2. \]
3. Van Kampen’s theorem gives\[ \pi_1(X_k) \cong \left\langle c_1,\dots,c_k,x_1,\dots,x_k \mathrel{\Big|} c_1\cdots c_k=1,\ c_i=x_i^2\ (1\le i\le k) \right\rangle. \]
Attach the Möbius bands one at a time, thickening each gluing circle by collars so that van Kampen applies to open sets. At the \(i\)th attachment, the intersection deformation retracts onto the common boundary circle. By 2, its generator maps to \(c_i\) on the \(S_k\) side and to \(x_i^2\) on the Möbius-band side. Van Kampen therefore adds the generator \(x_i\) and the relation \(c_i=x_i^2\). Doing this for all \(i\) yields the displayed presentation together with the relation from 1.
4. Eliminating the generators \(c_i\) gives\[ \boxed{ \pi_1(X_k) \cong \left\langle x_1,\dots,x_k \mathrel{\Big|} x_1^2x_2^2\cdots x_k^2=1 \right\rangle.} \]
Substitute \(c_i=x_i^2\) from 3 into the single relation \[ c_1c_2\cdots c_k=1. \] The generators \(c_i\) can then be removed by Tietze transformations, leaving exactly the displayed presentation.
5. The surface \(X_k\) is the closed nonorientable surface of genus \(k\):\[ \boxed{X_k\cong N_k=\#^k\RP^2.} \]
All \(k\) boundary circles of \(S_k\) are filled by Möbius bands, so \(X_k\) is a closed connected surface. It is nonorientable because it contains the core of any attached Möbius band with its Möbius neighborhood.
Moreover, \[ \chi(S_k)=2-k, \qquad \chi(M_i)=0, \qquad \chi(S^1)=0. \] Euler characteristic is unchanged when a Möbius band is glued along a boundary circle, so \[ \chi(X_k)=2-k. \] By the classification of closed connected surfaces, the unique nonorientable surface with Euler characteristic \(2-k\) is the connected sum of \(k\) projective planes. This also agrees with the standard presentation in 4.
Thus \(X_1\cong\RP^2\), \(X_2\) is the Klein bottle, and in general \(X_k\cong N_k\).