The problem is asking me to prove there exists a point in a compact connected surface that is not homeomorphic to the sphere, such that removing this point will give a space homotopic equivalent to a bouquet of circles. I haven't got any intuition to approach it. Could somebody offer a hint?
2026-03-26 06:20:28.1774506028
Introduction to Topological Manifolds Problem 7-14
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Suppose the surface is oriented You can draw the fundamental domain, it is a $2g$-gon, if you remove a point inside, the resulting space as a fundamental domain which retract to the boundary of the $2g$-gon, i.e the space retract to a bouquet of $2g$-circles.