Math 6510 • Differentiable Manifolds

The surface \(M_g\) of genus \(g\), embedded in \(\mathbb{R}^3\) in the standard way,
bounds a compact region \(R\). Two copies of \(R\), glued together by the identity map
between their boundary surfaces \(M_g\), form a closed 3-manifold \(X\). Compute the
homology groups of \(X\) via the Mayer-Vietoris sequence for this decomposition of \(X\)
into two copies of \(R\).

\emph{Diagram: a standard embedded genus \(g\) surface, drawn schematically as a handled
surface bounding a compact region \(R\).}