Calculating Euler-Poincare characteristic inductively

Milosav M. Marjanovic

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arXiv · 1212.0154

Calculating Euler-Poincare characteristic inductively

Milosav M. Marjanovic

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Abstract

Motivated by decompositions of spaces that arise in continuous and discrete Morse theory, we describe a so called fibrous decomposition Z = X_0(Y_1)X_1 ... X_{n-1}(Y_n)X_n of a space Z. Among the applications is a succinct formula for the Euler-Poincare characteristic of Z, e(Z) = e(X_0) - e(Y_1) + e(X_1) - ... + e(X_{n-1}) - e(Y_n) + e(X_n) which exhibits the familiar sign pattern. A substantial part of the paper are examples demonstrating how the fibrous decomposition and consequently the Euler-Poincare characteristic can be easily calculated without the use of any auxiliary combinatorial structure on spaces.

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Milosav M. Marjanovic. 2012-12-01. Calculating Euler-Poincare characteristic inductively. https://arxiv.org/abs/1212.0154

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