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Question

Math 6510 • Differentiable Manifolds

Summer 2018 • Question 10

Question

Math 6510 • Differentiable Manifolds Summer 2018 • Question 10

Let \(C_\bullet \) be a chain complex of finitely-generated \(\mathbb {Z}\)-modules such that \(C_i=0\) for all \(|i|\gg 0\) (i.e., the complex \(C_\bullet \) is bounded). Define the Euler characteristic of \(C\) to be \[ \chi (C)=\sum _{i\in \mathbb {Z}}(-1)^i\operatorname {rank}(C_i), \] where \(\operatorname {rank}(C_i)\) denotes the rank of the (finitely-generated) abelian group \(C_i\).

  1. Let \(X\) be a finite CW complex, i.e. one having finitely many cells in total. Define \(\chi (X)\) to be \(\chi (C_\bullet ^{\mathrm {CW}}(X))\), where \(C_\bullet ^{\mathrm {CW}}(X)\) denotes the cellular chain complex of \(X\). Show that \[ \chi (X)=\sum _{i\in \mathbb {Z}}(-1)^i\operatorname {rank}(H_i(X)), \] and that this quantity equals \(\sum _{i\in \mathbb {Z}}(-1)^i\dim _F(H_i(X;F))\) for any field \(F\).
  2. Let \(X\) be a connected compact manifold of odd dimension; you may assume \(X\) is homeomorphic to a finite CW complex. Show that \(\chi (X)=0\).

Raw LaTeX

Let \(C_\bullet\) be a chain complex of finitely-generated \(\Z\)-modules such
that \(C_i=0\) for all \(|i|\gg0\) (i.e., the complex \(C_\bullet\) is bounded).
Define the Euler characteristic of \(C\) to be
\[
  \chi(C)=\sum_{i\in\Z}(-1)^i\rk(C_i),
\]
where \(\rk(C_i)\) denotes the rank of the (finitely-generated) abelian group
\(C_i\).
\begin{enumerate}
\item Let \(X\) be a finite CW complex, i.e. one having finitely many cells in
total. Define \(\chi(X)\) to be \(\chi(C_\bullet^{\mathrm{CW}}(X))\), where
\(C_\bullet^{\mathrm{CW}}(X)\) denotes the cellular chain complex of \(X\). Show
that
\[
  \chi(X)=\sum_{i\in\Z}(-1)^i\rk(H_i(X)),
\]
and that this quantity equals
\(\sum_{i\in\Z}(-1)^i\dim_F(H_i(X;F))\) for any field \(F\).

\item Let \(X\) be a connected compact manifold of odd dimension; you may assume
\(X\) is homeomorphic to a finite CW complex. Show that \(\chi(X)=0\).
\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Chain Complexes and Chain Homotopy Euler Characteristic and Lefschetz Fixed Point Theorem Smooth Manifolds and Embedded Submanifolds