\chi = 2 - 2g
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- A mathematician working on the application of algebraic topology is analyzing a closed, orientable surface. Suppose the Euler characteristic of the surface is given by the formula \(\chi = 2 - 2g\), where \(g\) is the genus of the surface. If the surface is a double torus (i.e., \(g = 2\)), what is its Euler characteristic?
- To find the Euler characteristic of a double torus, we use the given formula:
- where \(g\) is the genus of the surface. For a double torus, \(g = 2\). Substituting this value into the formula, we get:
- \chi = 2 - 2 \times 2 = 2 - 4 = -2
- Therefore, the Euler characteristic of a double torus is \(\boxed{-2}\).
