Low-dimensional homology of finite Coxeter groups

We use the GAP package HAP to compute the integral homology up to degree 4 of various irreducible spherical Coxeter groups. There are 6 exceptional such groups and 4 infinite families. For types \(A_n\), \(B_n\) and \(D_n\), the homological stability theorem in Richard Hepworth's paper let's us extrapolate to the whole family from those examples we compute. For type \(I_2(p)\), namely the dihedral groups, the homology can be determined in general from work of David Handel, see the math stackexchange answer by Jim Belk. Thus we get the integral homology in degree up to 4 for all irreducible spherical Coxeter groups and thus all spherical Coxeter groups by the Künneth formula. Rachael Boyd's paper gives a formula for the integral homology in degrees 2 and 3 of an arbitrary finitely generated Coxeter group.

The results (requiring ~1 hour computation) are as follows. First, the values of \(H_1, H_2, H_3, H_4\) for the 6 exceptional groups:

\(H_*(W(F_4)) = \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{5} \oplus \mathbb{Z}_{3}^{2} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}^{7}\)

\(H_*(W(H_3)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{5}, \quad \mathbb{Z}_{2}^{2}\)

\(H_*(W(H_4)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4} \oplus \mathbb{Z}_{5}, \quad \mathbb{Z}_{2}^{2}\)

\(H_*(W(E_6)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}^{3}\)

\(H_*(W(E_7)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}^{5}\)

\(H_*(W(E_8)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}^{4}\)

For the examples in the infinite families:

\(H_*(W(A_1)) = \mathbb{Z}_{2}, \quad 0, \quad \mathbb{Z}_{2}, \quad 0\)

\(H_*(W(A_2)) = \mathbb{Z}_{2}, \quad 0, \quad \mathbb{Z}_{2} \oplus \mathbb{Z}_{3}, \quad 0\)

\(H_*(W(A_3)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}\)

\(H_*(W(A_4)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}\)

\(H_*(W(A_5)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}^{2}\)

\(H_*(W(A_6)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}^{2}\)

\(H_*(W(A_7)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}^{3}\)

\(H_*(W(B_2)) = \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}^{2}\)

\(H_*(W(B_3)) = \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{4} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}, \quad \mathbb{Z}_{2}^{5}\)

\(H_*(W(B_4)) = \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{3}, \quad \mathbb{Z}_{2}^{5} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{2}, \quad \mathbb{Z}_{2}^{9}\)

\(H_*(W(B_5)) = \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{3}, \quad \mathbb{Z}_{2}^{6} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{2}, \quad \mathbb{Z}_{2}^{12}\)

\(H_*(W(B_6)) = \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{3}, \quad \mathbb{Z}_{2}^{7} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{2}, \quad \mathbb{Z}_{2}^{14}\)

\(H_*(W(B_7)) = \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{3}, \quad \mathbb{Z}_{2}^{7} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{2}, \quad \mathbb{Z}_{2}^{15}\)

\(H_*(W(B_8)) = \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{3}, \quad \mathbb{Z}_{2}^{7} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{2}, \quad \mathbb{Z}_{2}^{16}\)

\(H_*(W(D_4)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{3}, \quad \mathbb{Z}_{2}^{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{3}, \quad \mathbb{Z}_{2}^{6}\)

\(H_*(W(D_5)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{2}, \quad \mathbb{Z}_{2}^{5}\)

\(H_*(W(D_6)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{4} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{2}, \quad \mathbb{Z}_{2}^{8}\)

\(H_*(W(D_7)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{2}, \quad \mathbb{Z}_{2}^{7}\)

\(H_*(W(D_8)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{2}, \quad \mathbb{Z}_{2}^{9}\)

\(H_*(W(D_9)) = \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}^{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}^{2}, \quad \mathbb{Z}_{2}^{8}\)

\(H_*(W(I_2(5))) = \mathbb{Z}_{2}, \quad 0, \quad \mathbb{Z}_{2} \oplus \mathbb{Z}_{5}, \quad 0\)

\(H_*(W(I_2(6))) = \mathbb{Z}_{2}^{2}, \quad \mathbb{Z}_{2}, \quad \mathbb{Z}_{2}^{3} \oplus \mathbb{Z}_{3}, \quad \mathbb{Z}_{2}^{2}\)