Draw a gluing diagram of the connected sum of two tori

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Reference no: EM131842947

Problem -

1. (a) (Problem 1.4(iii) in Sjamaar) Draw a gluing diagram of the connected sum of two tori. [Hint: I can imagine two di?erent ways to do this. One is to draw the connected sum of two tori and then imagine cutting it apart and laying it flat. The other is to take two copies of a gluing diagram for the torus with a hole punched in it and connect them in an appropriate way.]

(b) One of the gluing diagrams in Problem 1.5 of Sjamaar is the connected sum of two tori. By simplifying (if necessary) your answer to (a), find which one it is.

2. Consider the curve in R2 given by the equation y2 = x3 -x.

More generally, show that the curve given by the equation y2 = x3 + ax2 + bx (where a and b are constants) is a manifold if and only if a2b2 -4b3 6= 0.

Textbook - Manifolds and Differential Forms, Revised edition, 2017 by Reyer Sjamaar.

Reference no: EM131842947

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