WiSe 20/21 Algebraic Topology I Exercise sheet 8 (due January 13) Exercise 8.1. Consider a commutative diagram of abelian groups fn gn @n ··· An Bn Cn An−1 ··· an bn =∼ cn an−1 0 0 0 0 fn 0 gn 0 @n 0 ··· An Bn Cn An−1 ··· where the rows are long exact sequences and cn is an isomorphism for all n 2 Z. Verify that the associated \abstract Mayer{Vietoris" sequence 0 MV (an;−fn) 0 (fn;bn) 0 @n ··· An An ⊕ Bn Bn An−1 ··· MV −1 0 is exact, where @n = @n ◦ cn ◦ gn. Exercise 8.2. Let C and D be small categories, f; g : C ! D functors, and φ: f ) g a natural transformation. (a) Show that φ induces a homotopy between the morphisms of simplicial sets N(f);N(g): N(C) ! N(D) and of spaces jN(f)j; jN(g)j: jN(C)j ! jN(D)j: Hint. The natural transformation φ can be viewed as a functor C × [1] ! D. (b) Deduce that jN(−)j takes equivalences of categories to homotopy equivalences of spaces. (c) Suppose that C has either an initial object or a final object. Show that jN(C)j is contractible. Exercise 8.3. Let (Xα)α2A be a family of topological spaces with base points xα 2 Xα. Recall that their coproduct in Top∗ is given by the wedge sum ! _ a Xα = Xα =fxαgα2A: α2A α2A W W Let iβ : Xβ ,! α2A Xα be the canonical map and let pβ : α2A Xα Xβ be the pointed map such that pβ ◦ iα is the identity on Xβ if α = β and is constant otherwise. (a) If A is finite, show that the pointed map _ Y Xα ! Xα α2A α2A induced by the maps pβ ◦ iα : Xα ! Xβ is a homeomorphism onto its image. 1 (b) Suppose that, for every α 2 A, fxαg is closed in Xα and is a neighborhood deforma- tion retract, i.e., there exists a neighborhood Nα of xα such that idNα is homotopic rel fxαg to the constant map. Show that there an isomorphism ! ~ _ ∼ M ~ H∗ Xα = H∗(Xα) α2A α2A such that: ~ L ~ • the inclusion H∗(Xβ) ,! α2A H∗(Xα) is identified with (iβ)∗, L ~ ~ • the projection α2A H∗(Xα) H∗(Xβ) is identified with (pβ)∗. Exercise 8.4. Let n ≥ 1. We consider the n-sphere Sn ⊂ Rn+1 as a pointed topological space with base point e1 = (1; 0;:::; 0). Let n n n i1; i2 : S ,! S _ S be the summand inclusions and n n n p1; p2 : S _ S ! S the projections, as in Exercise 8.3. The fold map fold: Sn _ Sn ! Sn is the unique pointed map such that fold ◦ i1 = idSn and fold ◦ i2 = idSn . (a) Construct a pointed map pinch: Sn ! Sn _ Sn such that p1 ◦ pinch '∗ idSn and p2 ◦ pinch '∗ idSn . (b) Let f; g : Sn ! Sn be pointed maps. Show that n n f∗ + g∗ = (fold ◦ (f _ g) ◦ pinch)∗ : H∗(S ) ! H∗(S ): Hint. This is a formal consequence of (a) and Exercise 8.3(b). 2.