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Dave Witte Morris
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This problem is not research level. For $n \neq 1$, the given homeomorphisms act trivially on the fundamental group, but applying an orientation-reversing homeomorphism to the $S^1$-factor yields a homeomorphism that is not trivial on the fundamental group.

Dave Witte Morris
  • 3.9k
  • 1
  • 18
  • 20