Calculating Euler-Poincare characteristic inductively

2012/12/01 by Marjanovic, Milosav M.
#Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1212.0154

Abstract

Motivated by decompositions of spaces that arise in continuous and discrete Morse theory, we describe a so called fibrous decomposition Z = X₀(Y₁)X₁ ... 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(X₀) - e(Y₁) + e(X₁) - ... + 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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