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.
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The problem has been editted to require the condition for all x, not some, so the answer does apply.
Dave Witte Morris
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Dave Witte Morris
- 3.9k
- 1
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