Show that any compact connected surface with nonempty boundary is homotopy equivalent to a wedge of circles
Hint: you may assume that any compact connected surface without boundary is given by identifying edges of a polygon in pairs.
For each surface appearing in the classification of compact surfaces with nonempty boundary, say how many circles are needed in the wedge from part (a).
Hint: you should be able to do this even if you have not done part (a).
Solution
1. Every compact connected surface with nonempty boundary admits a handle decomposition with one \(0\)-handle, only \(1\)-handles, and no \(2\)-handles.
By the classification of compact surfaces with boundary, an orientable surface is obtained from a disk by adding \(g\) orientable handles and then creating \(b-1\) additional boundary components, while a nonorientable surface is obtained by adding \(h\) crosscaps and then creating \(b-1\) additional boundary components. These operations can be realized by attaching \(1\)-handles to a disk.
2. Such a surface deformation retracts onto the union of the core point of the \(0\)-handle and the core arcs of its \(1\)-handles, which is a wedge of circles.
A handlebody with only \(0\)- and \(1\)-handles collapses onto its \(1\)-dimensional spine. With one \(0\)-handle, the spine is a connected graph with one vertex and one loop for each \(1\)-handle.
3. For the orientable surface \(\Sigma_{g,b}\) of genus \(g\) with \(b\ge1\) boundary components,\[\boxed{\Sigma_{g,b}\simeq\bigvee^{\,2g+b-1}S^1.}\]
Its Euler characteristic is \(2-2g-b\). A wedge of \(r\) circles has Euler characteristic \(1-r\), so the spine in 2 must have \[r=1-\chi(\Sigma_{g,b})=2g+b-1\] loops.
4. For the nonorientable surface \(N_{h,b}\) with \(h\ge1\) crosscaps and \(b\ge1\) boundary components,\[\boxed{N_{h,b}\simeq\bigvee^{\,h+b-1}S^1.}\]
Here \(\chi(N_{h,b})=2-h-b\), so the same calculation gives \[r=1-\chi(N_{h,b})=h+b-1.\]
5. These formulas include the disk (\(r=0\)), annulus (\(r=1\)), Möbius band (\(r=1\)), punctured torus (\(r=2\)), and pair of pants (\(r=2\)).
Substitute the corresponding values of \((g,b)\) or \((h,b)\) into <1>3–<1>4.