M8130 Algebraic topology, tutorial 04, 2020
2. 11. 2020
Exercise 1. There is a lemma, that says: Given the following diagram, where rows are long exact sequences and m is an iso
Kn —-—> Ln —-—> Mn ——> Kn-i -> Ln-i -> Mn-i
i     i     mi     _i      _[ _[
Kn ——> Ln ——> Mn -> Kn-i -► Ln-i -> Mn_i
we get a long exact sequence
Kn    > Ln © Kn    y Ln      y  Kn—i    ^ ■ ■ ■
We can denote d = ho m_1 o j.
Show exactness in Ln © Kn and also in Ln.
M8130 Algebraic topology, tutorial 04, 2020
2. 11. 2020
Exercise 2. There is a long exact sequence of the triple (X,A,B), i.e. (B C A C X):
----> Hn(A, B)       Hn(X, B) ^ Hn(X, A) ^ Hn-i(A, B) -> ■ ■ ■ ,
d JA
with Hn(X,A) —Hn_i{A) —> Hn_i{A, B). We get this sequence from a special short exact sequence of chain complexes. Show that it is exact and that the triangle commutes, that is     = ja 0 <9*.
Exercise 3. Apply previous exercise to the triple (Dk, S*-1, *); where * is a point.
M8130 Algebraic topology, tutorial 04, 2020
2. 11. 2020
Exercise 4. Show that the chain in Ck(Ak ,dAk) given by id: Ak —> Ak is the representative of the generator of
Hk(Ak,dAk) 21 Z.
(Use induction and the long exact sequence for triple.)
M8130 Algebraic topology, tutorial 04, 2020
2. 11. 2020
Exercise 5. Using the Mayor-Vietoris exact sequence compute the homology groups of the torus.
M8130 Algebraic topology, tutorial 04, 2020
2. 11. 2020
Exercise 6. Prove Snake Lemma.
X
9(c)
1
rf) ^ i (h)l i V) = c.
M8130 Algebraic topology, tutorial 04, 2020 2. 11. 2020
Exercise 7. Prove 5-lemma.
&l     iA    ly    U U
Asu.m   $/ua#e LeiM.IM.Qs.