A degree-one proper map between non-compact surfaces has a geometric kernel or is properly homotopic to a homeomorphism, if it is injective on Brown's proper fundamental group at every end; a new cπ1 invariant yields further sufficient conditions.
$\pi_1$-injective proper maps between non-compact surfaces
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We classify all $\pi_1$-injective proper maps between non-compact surfaces up to proper homotopy.
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Geometric Kernels of Proper Maps Between Non-Compact Surfaces
A degree-one proper map between non-compact surfaces has a geometric kernel or is properly homotopic to a homeomorphism, if it is injective on Brown's proper fundamental group at every end; a new cπ1 invariant yields further sufficient conditions.