We describe how one can calculate the first and second rational (co)homology groups of the moduli spaces of smooth or stable n-pointed curves of any genus using only relatively simple algebraic geometry and Harer's vanishing theorem. In particular, we give a new proof of Harer's theorem describing the second cohomology group of the moduli space of n-pointed smooth curves of given genus. The idea is to use Deligne's Gysin spectral sequence, applied to the pair consisting of the moduli space of stable n-pointed curves and its Deligne-Mumford boundary. This is possible since, from the point of view of orbifolds, the boundary is a divisor with normal crossings in moduli. Roughly speaking, the Gysin spectral sequence calculates the cohomology of the open moduli space in terms of the cohomology of the strata of the stratification of the moduli space of stable n-pointed curves by "multiple intersections" of local components of the boundary. Knowing the first and second cohomology groups of the moduli spaces of stable n-pointed curves makes it possible to explicitly compute the low terms of the spectral sequence, and to conclude.
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