Algebraic topology - Classification of covering spaces Please write legibly (a) Let T = S S

 

Algebraic topology - Classification of covering spaces

Please write legibly

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(a) Let T² = S¹ × S¹ denote the 2-torus, and let K denote the Klein bottle. Show that T² is two-fold cover of K. (b) Can K be realized as a two-fold cover of itself? That is, does there exist a covering map p : K → K such that, for all x € K, the fiber p-¹(x) over x contains exactly 2 elements. (c) Find the universal covering of K. Justify your solution (i.e., show the covering space you obtain is simply connected). (a) Let T² = S¹ × S¹ denote the 2-torus, and let K denote the Klein bottle. Show that T² is two-fold cover of K. (b) Can K be realized as a two-fold cover of itself? That is, does there exist a covering map p : K → K such that, for all x € K, the fiber p-¹(x) over x contains exactly 2 elements. (c) Find the universal covering of K. Justify your solution (i.e., show the covering space you obtain is simply connected).

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