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Question

Math 6510 • Differentiable Manifolds

Fall 2015 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 2015 • Question 5

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3\mid x^2+y^2+z^2\}\). Define \(\pi :\mathbb {R}^3\backslash \{0\}\to S^2\) by \[ \pi (x,y,z)= \left ( \frac {x}{\sqrt {x^2+y^2+z^2}}, \frac {y}{\sqrt {x^2+y^2+z^2}}, \frac {z}{\sqrt {x^2+y^2+z^2}} \right ) \] and \(T_\epsilon (x,y,z)=(x,y,z+\epsilon )\). For \(\epsilon \in (0,1)\) the map \(f_\epsilon =\pi \circ T_\epsilon \) is a Lefschetz map from \(S^2\) to itself. Calculate its Lefschetz number and conclude every map of \(S^2\) to itself that is homotopic to the identity has a fixed point. (Hint: It will be easier to calculate the derivative of \(\pi \) and \(T_\epsilon \) separately and use the chain rule to find the derivative of \(f_\epsilon \).)

Let \(S^2=\{(x,y,z)\in\R^3\mid x^2+y^2+z^2\}\). Define
\(\pi:\R^3\backslash\{0\}\to S^2\) by
\[
  \pi(x,y,z)=
  \left(
    \frac{x}{\sqrt{x^2+y^2+z^2}},
    \frac{y}{\sqrt{x^2+y^2+z^2}},
    \frac{z}{\sqrt{x^2+y^2+z^2}}
  \right)
\]
and \(T_\epsilon(x,y,z)=(x,y,z+\epsilon)\). For \(\epsilon\in(0,1)\) the map
\(f_\epsilon=\pi\circ T_\epsilon\) is a Lefschetz map from \(S^2\) to itself.
Calculate its Lefschetz number and conclude every map of \(S^2\) to itself that is
homotopic to the identity has a fixed point. (Hint: It will be easier to calculate
the derivative of \(\pi\) and \(T_\epsilon\) separately and use the chain rule to find
the derivative of \(f_\epsilon\).)

Topic Tags

Chain Complexes and Chain Homotopy Euler Characteristic and Lefschetz Fixed Point Theorem Fixed Point Theorems and Borsuk-Ulam Homotopy Equivalence, Retracts, and Contractibility