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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1994 • Question 4

Question

Math 6510 • Differentiable Manifolds Fall 1994 • Question 4

Let \(K\) denote a finite simplicial complex. The Euler characteristic is given by \[ \chi (K)=\sum _{j\ge 0}(-1)^j\alpha _j \] where \(\alpha _j\) is the number of \(j\)-simplices in \(K\). If \(\mathrm {Sd}\,K\) denotes barycentric subdivision then \(\chi (K)=\chi (\mathrm {Sd}\,K)\).

Raw LaTeX

Let \(K\) denote a finite simplicial complex. The Euler
characteristic is given by
\[
  \chi(K)=\sum_{j\ge 0}(-1)^j\alpha_j
\]
where \(\alpha_j\) is the number of \(j\)-simplices in \(K\). If \(\mathrm{Sd}\,K\) denotes
barycentric subdivision then \(\chi(K)=\chi(\mathrm{Sd}\,K)\).

Topic Tags

Euler Characteristic and Lefschetz Fixed Point Theorem Simplicial and Delta Complexes