Fundamental Group & Covering Spaces

Question 1

deck transformation

Answer 1

homeomorphism φ from E to E such that pφ=p

Question 2

universal cover

Answer 2

E is simply connected and there is covering map p from E to X

Question 3

semi-locally simply connected

Answer 3

each point in X has a neighbourhood U with the property that any loop in U is based homotopic in X to the constant loop

Question 4

Borsuk-Ulam Theorem

Answer 4

for any integers n>m>=0 there exists no continuous map φ from S^n to S^m which is antipode preserving

Question 5

Ham Sandwich Theorem

Answer 5

Let X_1,…,X_n be bounded Lebesque measurable subsets of R^n. There exists a hyperplane bisecting each X_i simultaneously

Question 6

continuous action

Answer 6

the map to the action is continuous

Question 7

covering space action

Answer 7

continuous and every point in X has an open neighbourhood S such that gS intersect with S is empty for all non trivial g