In a computation analogous to Examples 44. 9 and 44.12 of the text, find the relative homology groups H_n( X, a) for the torus X with subcomplex a, as shown in Figs. 42.13 and Fig. 42.14.

Data from in Example 44.9

Let X be the simplicial complex consisting of the edges (excluding the inside) of the triangle in Fig. 44.10, and let Y be the subcomplex consisting of the edge P₂P₃. We have seen that H₁(X) ≈ H₁(S¹) ≈ Z. Shrinking P₂P₃ to a point collapses the rim of the triangle, as shown in Fig. 44.11. The result is still topologically the same as S¹. Thus, we would expect again to have H₁(X, Y) ≈ Z. Generators for C₁(X) are P₁P₂, P₂P₃, and P₃P₁. Since P₂P₃ ∈ C₁(Y), we see that generators of C₁(X)/C₁(Y) are  P₁P₂ + C₁(Y) and P₃P₁ + C₁(Y). To find Z₁( X, Y) we compute 

since P₂, P₃ ∈ C₀(Y). Thus for a cycle, we must have m = n, so a generator of Z₁(X, Y) is (P₁P₂ + P₃P₁) + C₁(Y). Since B₁(X, Y) = 0, we see that indeed H₁(X, Y) ≈ Z. Since P₁ + C₀(Y) generates Z₀(X, Y) and ∂₁(P₂P₁ + C₁(Y)) = (P₁ - P₂) + C₀(Y) =P₁ + C₀(Y), we see that H₀(X, Y) = 0. This is characteristic of relative homology groups of dimension 0 for connected simplicial complexes.

Transcribed Image Text:

J₁ (n P₁ P2 +mP3 P₁ + C₁(Y)) = ₁(n P₁ P₂) + 0₁ (m P3 P₁) + Co(Y) = n(P₂ - P₁) +m(P₁ - P3) + Co(Y) (mn) P₁+ Co(Y),