Part II | Algebraic Topology | Year 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2021 5 Paper 1, Section II 21F Algebraic Topology (a) What does it mean for two spaces X and Y to be homotopy equivalent? (b) What does it mean for a subspace Y X to be a retract of a space X? What ⊆ does it mean for a space X to be contractible? Show that a retract of a contractible space is contractible. (c) Let X be a space and A X a subspace. We say the pair (X, A) has the ⊆ homotopy extension property if, for any pair of maps f : X 0 Y and H : A I Y ×{ } → 0 × → with f A 0 = H0 A 0 , | ×{ } | ×{ } there exists a map H : X I Y with × → H X 0 = f, H A I = H0. | ×{ } | × Now suppose that A X is contractible. Denote by X/A the quotient of X by the ⊆ equivalence relation x x if and only if x = x or x, x A. Show that, if (X, A) satisfies ∼ 0 0 0 ∈ the homotopy extension property, then X and X/A are homotopy equivalent. Part II, 2021 List of Questions [TURN OVER] 2021 6 Paper 2, Section II 21F Algebraic Topology (a) State a suitable version of the Seifert–van Kampen theorem and use it to calculate the fundamental groups of the torus T 2 := S1 S1 and of the real projective 2 × plane RP . 2 2 (b) Show that there are no covering maps T 2 RP or RP T 2. → → (c) Consider the following covering space of S1 S1: ∨ a a b b b b a a Here the line segments labelled a and b are mapped to the two different copies of S1 contained in S1 S1, with orientations as indicated. ∨ Using the Galois correspondence with basepoints, identify a subgroup of π (S1 S1, x ) = F 1 ∨ 0 2 (where x0 is the wedge point) that corresponds to this covering space. Part II, 2021 List of Questions 2021 7 Paper 3, Section II 20F Algebraic Topology Let X be a space. We define the cone of X to be CX := (X I)/ × ∼ where (x , t ) (x , t ) if and only if either t = t = 1 or (x , t ) = (x , t ). 1 1 ∼ 2 2 1 2 1 1 2 2 (a) Show that if X is triangulable, so is CX. Calculate Hi(CX). [You may use any results proved in the course.] (b) Let K be a simplicial complex and L K a subcomplex. Let X = K , A = L , ⊆ | | | | and let X be the space obtained by identifying L K with L 0 C L . Show 0 | | ⊆ | | | | × { } ⊆ | | that there is a long exact sequence Hi+1(X0) Hi(A) Hi(X) Hi(X0) Hi 1(A) · · · → → → → → − → · · · H1(X0) H0(A) Z H0(X) H0(X0) 0. · · · → → → ⊕ → → (c) In part (b), suppose that X = S1 S1 and A = S1 x X for some x S1. × × { } ⊆ ∈ Calculate Hi(X0) for all i. Paper 4, Section II 21F Algebraic Topology (a) Define the Euler characteristic of a triangulable space X. 2 (b) Let Σg be an orientable surface of genus g. A map π :Σg S is a double- 2 → branched cover if there is a set Q = p1, . , pn S of branch points, such that the 1 2 { } ⊆ restriction π :Σg π− (Q) S Q is a covering map of degree 2, but for each p Q, 1 \ → \ 2 ∈ π− (p) consists of one point. By carefully choosing a triangulation of S , use the Euler characteristic to find a formula relating g and n. Part II, 2021 List of Questions [TURN OVER] 2020 5 Paper 1, Section II 21F Algebraic Topology Let p : R2 S1 S1 =: X be the map given by → × 2πir1 2πir2 p(r1, r2) = e , e , 1 where S is identified with the unit circle in C. [You may take as given that p is a covering map.] 2 (a) Using the covering map p, show that π1(X, x0) is isomorphic to Z as a group, where x = (1, 1) X. 0 ∈ (b) Let GL2(Z) denote the group of 2 2 matrices A with integer entries such that × 2 2 det A = 1. If A GL2(Z), we obtain a linear transformation A : R R . Show that ± ∈ → this linear transformation induces a homeomorphism fA : X X with fA(x0) = x0 and → 2 2 such that fA : π1(X, x0) π1(X, x0) agrees with A as a map Z Z . ∗ → → (c) Let p : X X for i = 1, 2 be connected covering maps of degree 2. Show that i i → there exist homeomorphisms φ : X X and ψ : X X so that the diagram 1 → 2 → b φ b Xb1 / X2 p1 p2 b b X / X ψ is commutative. Paper 2, Section II 21F Algebraic Topology (a) Let f : X Y be a map of spaces. We define the mapping cylinder M of f to → f be the space (([0, 1] X) Y )/ × t ∼ with (0, x) f(x). Show carefully that the canonical inclusion Y, M is a homotopy ∼ → f equivalence. (b) Using the Seifert–van Kampen theorem, show that if X is path-connected and α : S1 X is a map, and x = α(θ ) for some point θ S1, then → 0 0 0 ∈ π (X D2, x ) = π (X, x )/ [α] . 1 ∪α 0 ∼ 1 0 hh ii Use this fact to construct a connected space X with π (X) = a, b a3 = b7 . 1 ∼ h | i (c) Using a covering space of S1 S1, give explicit generators of a subgroup of F ∨ 2 isomorphic to F3. Here Fn denotes the free group on n generators. Part II, 2020 List of Questions [TURN OVER] 2020 6 Paper 3, Section II 20F Algebraic Topology Let K be a simplicial complex with four vertices v , . , v with simplices v , v , v , 1 4 h 1 2 3i v , v and v , v and their faces. h 1 4i h 2 4i (a) Draw a picture of K , labelling the vertices. | | (b) Using the definition of homology, calculate Hn(K) for all n. (c) Let L be the subcomplex of K consisting of the vertices v1, v2, v4 and the 1- simplices v , v , v , v , v , v . Let i : L K be the inclusion. Construct a simplicial h 1 2i h 1 4i h 2 4i → map j : K L such that the topological realisation j of j is a homotopy inverse to i . → | | | | Construct an explicit chain homotopy h : C (K) C (K) between i j and idC (K), • → • • ◦ • • and verify that h is a chain homotopy. Paper 4, Section II 21F Algebraic Topology In this question, you may assume all spaces involved are triangulable. (a) (i) State and prove the Mayer–Vietoris theorem. [You may assume the theorem that states that a short exact sequence of chain complexes gives rise to a long exact sequence of homology groups.] (ii) Use Mayer–Vietoris to calculate the homology groups of an oriented surface of genus g. (b) Let S be an oriented surface of genus g, and let D1,...,Dn be a collection of mutually disjoint closed subsets of S with each Di homeomorphic to a two-dimensional disk. Let Di◦ denote the interior of Di, homeomorphic to an open two-dimensional disk, and let T := S (D◦ D◦ ). \ 1 ∪ · · · ∪ n Show that Z i = 0, 2g+n 1 Hi(T ) = Z − i = 1, 0 otherwise. (c) Let T be the surface given in (b) when S = S2 and n = 3. Let f : T S1 S1 → × be a map. Does there exist a map g : S1 S1 T such that f g is homotopic to the × → ◦ identity map? Justify your answer. Part II, 2020 List of Questions 2019 6 Paper 3, Section II 20F Algebraic Topology Let K be a simplicial complex, and L a subcomplex. As usual, Ck(K) denotes the group of k-chains of K, and Ck(L) denotes the group of k-chains of L. (a) Let Ck(K,L)= Ck(K)/Ck(L) for each integer k. Prove that the boundary map of K descends to give C (K,L) the • structure of a chain complex. (b) The homology groups of K relative to L, denoted by Hk(K,L), are defined to be the homology groups of the chain complex C (K,L). Prove that there is a long exact • sequence that relates the homology groups of K relative to L to the homology groups of K and the homology groups of L. (c) Let D be the closed n-dimensional disc, and Sn 1 be the (n 1)-dimensional n − − sphere. Exhibit simplicial complexes Kn and subcomplexes Ln 1 such that Dn ∼= Kn in n 1 − | | such a way that Ln 1 is identified with S − . | − | (d) Compute the relative homology groups Hk(Kn,Ln 1), for all integers k > 0 and − n > 2 where Kn and Ln 1 are as in (c). − Paper 4, Section II 21F Algebraic Topology State the Lefschetz fixed point theorem. Let n 2 be an integer, and x S2 a choice of base point. Define a space > 0 ∈ 2 X := (S Z/nZ)/ × ∼ where Z/nZ is discrete and is the smallest equivalence relation such that (x0, i) ∼ ∼ ( x0, i + 1) for all i Z/nZ. Let φ : X X be a homeomorphism without fixed points. − ∈ → Use the Lefschetz fixed point theorem to prove the following facts. 3 (i) If φ = IdX then n is divisible by 3.