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I am reading Guillemin and Pollack.

The definition of an immersion they give is:

$f: X \to Y$ is an immersion at $x \in X$ if $df_x : T_x(X) \to T_y(Y)$ is injective. If $f$ is an immersion $\forall x \in X$, then we say that $f$ is an immersion.

So apparently the map from the circle to the figure 8 is an immersion, as they state on the next page. But what about the critical point in the mapping $f$? There is no tangent space defined here, correct? So then how could $f$ possibly be an immersion?

Also, is there a simple example of something that isn't an immersion that will help me remember this definition?

Thank you.

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    $\begingroup$ Consider the figure eight as a subspace of $\mathbb{R}^2$. Then $f \colon S^1 \to \mathbb{R}^2$ is an immersion [if $f$ is chosen correctly] with image "$8$". $\endgroup$ – Daniel Fischer Sep 8 '16 at 18:32
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    $\begingroup$ The map is from $S^1 \to \Bbb R^2$. You need to check (after suitably defining a parametrization), that the derivative $df_x : T_xS^1 \to T_{f(x)}\Bbb R^2$ is injective at all $x$ in $S^1$. $\endgroup$ – PVAL-inactive Sep 8 '16 at 18:33
  • $\begingroup$ Which point do you think is critical? $\endgroup$ – Anthony Carapetis Sep 9 '16 at 3:58
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you can consider the derivative map of $f$ such a way that map a tangent vector on $S^{1}$ to a tangent vector on eight figure. Hence each tangent vector maps to only one tangent vector on eight figure. So $df_{x}$ is one-to-one.

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