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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1994 • Question 11

Question

Math 6510 • Differentiable Manifolds Fall 1994 • Question 11

Let \(\Sigma \) denote the surface obtained from an annulus by identifying antipodal points of the inner circle and by identifying antipodal points of the outer circle. Compute the fundamental group of \(\Sigma \). Can you identify \(\Sigma \) as a connected sum of tori and projective planes?

Raw LaTeX

Let \(\Sigma\) denote the surface obtained from an annulus by
identifying antipodal points of the inner circle and by identifying antipodal points
of the outer circle. Compute the fundamental group of \(\Sigma\). Can you identify
\(\Sigma\) as a connected sum of tori and projective planes?

Topic Tags

Fundamental Groups Real Projective Spaces Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups