Will every continuous map from $S^1$ to itself have a fixed point?

Question

Will every continuous map from $S^1$ to itself have a fixed point? I cant understand how to conclude anything from this

@learnmore: Rotate the circle through a right angle around its centre. Is that a continuous map? Does it have a fixed point? — Brian M. Scott, Nov 22 '14 at 07:02

score 0 · Answer 1 · answered Nov 22 '14 at 07:09

To expand on what's been said in the comments: consider the unit circle with polar coordinates; $\mathbb{S}^1 = \{ (1, \theta) : \theta \in [0,2 \pi) \}$. Then the map

$f(1,\theta) = (1, \theta + \frac{\pi}{2} \hspace{0.3cm} \text{mod} \hspace{0.3cm} 2 \pi)$

is a rotation. It's continuous but doesn't have a fixed point.

Will every continuous map from $S^1$ to itself have a fixed point?

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