Results on twisted homology of complexes of dimension two, the plus construction, and Wall's D(2) problem

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Twisted homology in degree zero — Lemma 4.2. For a connected CW complex Z and a nontrivial character rho of its fundamental group, H_0(Z; rho) = 0.
• The Euler characteristic with twisted coefficients — Lemma 4.3. For a finite connected complex of dimension at most two and every character rho: dim H_0 - dim H_1 + dim H_2 of the homology twisted by rho is the Euler characteristic.
• The boundary of a presentation complex by Fox derivatives — Lemma 5.1. For the presentation complex B of <x_1..x_g | r_1..r_h> and a character rho: the boundary C_2(B; rho) -> C_1(B; rho) has the matrix of the Fox derivatives of the relators evaluated at rho, and H_2(B; rho) = 0 exactly if its h rows are linearly independent. Uses TOP1 to TOP3, which are proved in the programme.
• Twisted second homology of infinite cyclic covers — Lemma 1.1. For a finite connected complex A of dimension at most two, a character rho and an infinite cyclic cover: if H_2(A; rho) = 0 then the second homology of the cover twisted by the restricted character is 0.
• The tower obstruction — Theorem 2.1. Assume the homotopy invariance in TOP5. Let Y be a finite connected CW complex of dimension at most two and rho a character of its fundamental group with H_2(Y; rho) = 0. For every map from a finite connected simplicial complex F with first Betti number 0 to Y, the pullback of rho is trivial.
• Characters and elements of finite order — Corollary 3.1. Assume the homotopy invariance in TOP5. If Y is a finite connected CW complex of dimension at most two and H_2(Y; rho) = 0 for a character rho, then rho(g) = 1 for every element g of finite order of the fundamental group.
• The finite plus construction — Proposition 3.1. Let B be a finite connected complex of dimension at most two and K the normal closure of t_1, ..., t_r in P = pi_1(B), with K perfect, G = P/K. Then B is contained in a finite complex X with r more cells of dimension two and r cells of dimension three such that pi_1(X) = G, H_i of the universal cover vanishes for i > 2, H^3(X; M) = 0 for every Z[G]-module M, and H_*(B; N) = H_*(X; N) for every right Z[G]-module N.
• A perfect normal subgroup — Lemma 1.1. In the group P = <x_1, a_1, x_2, a_2, s | x_1 a_1 x_1^-1 = a_1^4, x_2 a_2 x_2^-1 = a_2^3, x_1 = a_2^13, s x_2 s^-1 = a_1^5> the normal closure of x_1^2 and x_2^3 is perfect.
• The matrix of Fox derivatives of the example — Lemma 3.1. For the character rho(x_1) = -1, rho(a_1) = zeta, rho(x_2) = zeta^2, rho(a_2) = -1, rho(s) = 1 (zeta a primitive cube root of unity) the four rows of Fox derivatives of the relators of P are linearly independent; the minor on the columns x_1, a_1, x_2, s is -4 (1 - zeta^2). So H_2(B; rho) = 0 for the presentation complex B.
• A counterexample to the finite D(2) problem — Theorem 4.1. Assume the homotopy invariance in TOP5. There is a finite connected CW complex X of dimension three (one vertex, five edges, six two-cells, two three-cells) with H_i of the universal cover 0 for i > 2 and H^3(X; M) = 0 for every module M over the group ring of its fundamental group, that is not homotopy equivalent to any finite CW complex of dimension at most two.

Reading order

Four lessons, to be read in order. The first lesson lists six standard theorems of algebraic topology (cells and the fundamental group, covering spaces, cellular homology, the Hurewicz theorem, coefficients in modules over the group ring, simplicial models); all are proved in the programme, in the course on cell complexes, fundamental groups and covering spaces and in the course on characteristic classes, except the homotopy invariance of homology with coefficients in modules over the group ring, a named hypothesis with a place in a free textbook. Everything else is proved: the tower obstruction, the plus construction, the perfect subgroup and the matrix computation. The theorem is proved relative to that one hypothesis.