HW1.
Home work due Sept.20.
1. Show that if {C∗ , d} is acylic, then there is a chain map φ : {0, 0} → {C∗ , d} where {0, 0}
is the zero chain complex and φ is a quasi-isomorphism.
2. Let Γ be a connected graph with V nodes and E edges. Compute H0 (Γ), H1 (Γ), and the
Euler number χ(Γ).
3. Let K be the simplicial complex with vertices a < b < c and edges ab, ac, bc. Compute the
cycles Z1 (K) and the homology H1 (K).
4. Let L be the simplicial complex with vertices A < B < C < D, edges AB, AC, AD, BC, BD, CD
and faces ABC, ABD, ACD, BCD. Compute H0 (L), H1 (L), and H2 (L). Describe the generator of
H2 (L) as a chain.
5. Find a simplicial description of the 2-torus T and use it to compute H0 (T ), H1 (T ), and H2 (T ).
Describe the generators of H1 (T ) and H2 (T ) as chains.
6. Give a description of the 2-torus T as a ∆-set, Use this description to compute H0 (T ), H1 (T ),
and H2 (T ). Describe the generators of H1 (T ) and H2 (T ) as chains.
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