Problems

Part 1

Let M be the M¨obius Strip obtained from identifying (0, t) with (1, 1 – t) on

the square I × I. Let A be the boundary circle i.e. the image of I × {0, 1}

under the quotient map. Recall that using the long exact sequence of the pair

(M, A), the frst relative homology H1(M, A) ∼ = Z/2Z. Prove using the long

exact sequence of the triple (M, B, A) that the curve f(t) = (0, t) (thought of

as a 1-chain) represents the non-zero class in this group. Here B is the image

of the boundary of I × I in M.

Part 2

Let (C•, ∂) and (D•, ∂) be chain complexes. Show that there is a chain complex

hom• with

homn = Y

j–i=n

Hom(Ci, Dj)

so that the space of cycles Z0(hom•) = ker(∂ : hom0 → hom–1) is the space

of chain maps C• → D•, and the 0th homology H0(hom•) is the space of chain

maps modulo chain homotopies.

Notation: For abelian groups G1, G2, the set Hom(G1, G2) of group homomorphisms G1 → G2 is itself an abelian group under the operation (ϕ, ψ) 7→ ϕ + ψ

where (ϕ + ψ)(g) = ϕ(g) + ψ(g).

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