Calculating Euler-Poincare characteristic inductively

TL;DR

This paper introduces a fibrous decomposition method for spaces that simplifies calculating the Euler-Poincare characteristic, providing a recursive formula and practical examples without auxiliary structures.

Contribution

It presents a novel fibrous decomposition framework that yields a straightforward formula for Euler-Poincare characteristic calculations.

Findings

The formula e(Z) = e(X_0) - e(Y_1) + ... + e(X_n) accurately computes Euler-Poincare characteristics.

Examples demonstrate ease of calculation without auxiliary combinatorial structures.

The method applies to spaces arising in Morse theory, both continuous and discrete.

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.