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Mar 30, 2023 at 20:02 comment added Rafi Ps.: Let X be the genus 2 surface. There is an obvious continuous map f : X → S¹ ∨ S¹ that induces a surjection on π₁ (for a suitable choice of basepoint). Pulling back along f then gives us an example of a noninvertible endomorphism of a connected cover of a compact smooth surface. So such (endomorphisms of) covers exist for even the most "real world" examples of spaces.
S Mar 29, 2023 at 21:13 review First answers
Mar 29, 2023 at 21:46
S Mar 29, 2023 at 21:13 history edited Rafi CC BY-SA 4.0
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S Mar 29, 2023 at 20:44 review First answers
Mar 29, 2023 at 20:54
S Mar 29, 2023 at 20:44 history answered Rafi CC BY-SA 4.0