Calculating Euler-Poincare characteristic inductively
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 ... Xn-1(Yn)Xn 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(Xn-1) - e(Yn) + e(Xn) 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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