Find a way of identifying pairs of faces of \(\Delta^3\) to produce a \(\Delta\)-complex structure on \(S^3\) having a single 3 simplex, and compute the simplicial homology groups of this \(\Delta\)-complex.
Solution
Goal. Give \(S^3\) a \(\Delta\)-complex structure with a single \(3\)-simplex and compute its simplicial homology.
1. Construct the \(\Delta\)-complex by identifying the four faces of \(\Delta^3\) in two pairs.
1.1. Label the vertices \(v_0, v_1, v_2, v_3\) and the faces \(F_0 = [v_1,v_2,v_3]\), \(F_1 = [v_0,v_2,v_3]\), \(F_2 = [v_0,v_1,v_3]\), \(F_3 = [v_0,v_1,v_2]\).
these are the four \(2\)-faces of the tetrahedron \(\Delta^3\).
1.2. Identify \(F_0 \sim F_1\) (sending \(v_1 \mapsto v_0\)) and \(F_2 \sim F_3\) (sending \(v_3 \mapsto v_2\)).
this glues the four faces in two pairs, as required.
1.3. The resulting \(\Delta\)-complex has one \(3\)-simplex, two \(2\)-simplices, three \(1\)-simplices, and two \(0\)-simplices.
the two face-pairs give two \(2\)-simplices \(P = F_0 \sim F_1\) and \(Q = F_2 \sim F_3\); the vertex identifications give \(v_0 \sim v_1 =: A\) and \(v_2 \sim v_3 =: B\); the six edges collapse to three: the loop \(e_{01} = [v_0,v_1]\) at \(A\), the loop \(e_{23} = [v_2,v_3]\) at \(B\), and a single edge \(c\) from \(A\) to \(B\) (the four cross-edges \([v_0,v_2], [v_0,v_3], [v_1,v_2], [v_1,v_3]\) are all identified).
2. Compute the boundary maps.
2.1. \(\partial_3 = 0\).
\(\partial_3(\sigma) = F_0 - F_1 + F_2 - F_3 = P - P + Q - Q = 0\).
2.2. \(\partial_2(P) = e_{23}\) and \(\partial_2(Q) = e_{01}\).
\(\partial_2(F_0) = [v_2,v_3] - [v_1,v_3] + [v_1,v_2] = e_{23} - c + c = e_{23}\); \(\partial_2(F_2) = [v_1,v_3] - [v_0,v_3] + [v_0,v_1] = c - c + e_{01} = e_{01}\).
2.3. \(\partial_1(e_{01}) = 0\), \(\partial_1(e_{23}) = 0\), \(\partial_1(c) = B - A\).
\(e_{01}\) is a loop at \(A\) and \(e_{23}\) a loop at \(B\), so their boundaries vanish; \(c\) runs from \(A\) to \(B\).
3. Compute the homology.
3.1. \(H_0 = \ZZ\).
\(C_0 = \ZZ^2\) and \(\im \partial_1 = \ZZ(B - A)\), so \(H_0 = \ZZ^2/\ZZ = \ZZ\) (the space is connected).
3.2. \(H_1 = 0\).
\(\ker \partial_1 = \ZZ e_{01} \oplus \ZZ e_{23} = \ZZ^2\), and \(\im \partial_2 = \ZZ e_{23} \oplus \ZZ e_{01} = \ZZ^2\), so \(H_1 = \ZZ^2/\ZZ^2 = 0\).
3.3. \(H_2 = 0\).
\(\partial_2\) is injective (it sends the basis \(P, Q\) to the independent elements \(e_{23}, e_{01}\)), so \(\ker \partial_2 = 0\); hence \(H_2 = 0/\im \partial_3 = 0\).
3.4. \(H_3 = \ZZ\).
\(\ker \partial_3 = \ZZ\) and there is no \(4\)-simplex, so \(H_3 = \ZZ/0 = \ZZ\).
4. Q.E.D.
\(H_0 = H_3 = \ZZ\) and \(H_1 = H_2 = 0\), the homology of \(S^3\).