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A little category theory, the fundamental group, covering spaces Omar Antolin Camarena Due [Wed Jan 18]. The goals of this assignment are to give you some familiarity with categories and groupoids, and to “review” the fundamental group and covering space theory. No thought was given to the order of the problems, so there’s only around a 0.2% chance they are ordered by difficulty. Groups as categories Given any group G we can form a corresponding category BG as follows: • BG has only one object, say ∗. • The set of morphisms in BG from ∗ to ∗ is defined to be Hom(∗, ∗) := G, the set of elements of the group G. • The composition law Hom(∗, ∗) × Hom(∗, ∗) → Hom(∗, ∗) is the multiplication map G × G → G. 1. Given two groups G and H, one can form the functor category1 Fun(BG, BH). a) Describe Fun(BG, BH) in terms of concepts of group theory. b) Give an example of G 6= 1 and H 6= 1 such that Fun(BG, BH) is BK for some other group K. c) Show that if H is abelian, Fun(BG, BH) is isomorphic to BH × C for some other category C . 2. The “free loop category” of a category C is defined2 to be LC := Fun(BZ, C ). F a) Prove that for a group G, LBG is equivalent to a disjoint union g∈R BCG (g) where R is a set of representatives for the conjugacy classes of G and CG (g) = { h ∈ G : gh = hg} is the centralizer of g. b) Find a non-trivial category C such that LC is equivalent to C . Here “non-trivial” means containing non-identity morphisms. c) Find a formula for the n-th free loop category of BS 3 , up to equivalence. That is, find a simple explicit description of a category that is equivalent to L n BS 3 := L(L(· · · L(BS 3 ))). 1 If you’re not familiar with functor categories, consider looking it up as part of the assignment. Alternatively, you can just guess how the composition works after I tell you that if C and D are two categories then the objects of Fun(C , D ) are functors C → D , and the morphisms are natural transformations. 2 For purposes of this assignment! This isn’t a standard definition. 1 Fundamental group and covering spaces 3. Think of S 2 as the space of unit length vectors in R3 . Let f : S 2 → R be a continuous function. Prove that there exist three mutually orthogonal unit vectors u1 , u2 , u3 such that f (u1 ) = f (u2 ) = f (u3 ). 4. Find two compact connected surfaces that are not homotopy equivalent but become homotopy equivalent after removing one point from each. 5. Let p be a prime number congruent to 1 modulo 4 and let P p be the (geometric realization of the) clique complex of the Paley graph for p. You don’t need to know the definitions of any of those terms, as an explicit description of the space P p follows: Let e 1 , . . . , e p be the standard basis of R p and for any set S ⊆ {1, . . . , p} let E(S) be the convex P P S hull of { e s : s ∈ S }, that is, let E(S) := { s∈S λs e s : λs ≥ 0, s∈S λs = 1}. Then P p := S E(S) where the union runs over all S such that for every s, t ∈ S, s − t is a square modulo p (that is, such that there exists an integer r with r 2 ≡ s − t (mod p)). Compute the fundamental group of as many P p as you can, at least of P 5 , P 13 , and P 17 . 6. How many index 2 subgroups does the free group on 2 generators have? Use graphs to illustrate your answer. 7. Some changes were made to this question: part b) now says “automorphisms” instead of “endomorphisms”, part a) was changed and made optional. Let p : Y → X be a covering map with X and Y path-connected. Recall that a continuous function f : Y → Y such that p ◦ f = p is called an endomorphism of p and that if f has an inverse endomorphism, it is called an automorphism. a) [Two optional questions for extra credit] • If the fundamental group of Y is finite, show that any such endomorphism is automatically an automorphism. • Give an example of an endomorphism of a connected cover Y which is not an automorphism. b) Describe the group of automorphisms of p in terms of G := π1 (X , p(y0 )) and H := p ∗ (π1 (Y , y0 )). 2