Cohomology of Split Group Extensions and Characteristic Classes

There are characteristic classes that are the obstructions to the vanishing of the differentials in the Lyndon-Hochischild-Serre spectral sequence of an extension of an integral lattice L by a group G. These characteristic classes exist in a given page of the spectral sequence provided the differentials in the previous pages are all zero. When L decomposes into a sum of G-sublattices, we show that there are defining relations between the characteristic classes of L and the characteristic classes of its summands.

💡 Research Summary

The paper investigates the cohomology of split extensions of an integral lattice L by a group G, focusing on the obstruction classes—called characteristic classes—that control the differentials in the Lyndon‑Hochschild‑Serre (LHS) spectral sequence. For a split exact sequence
(1\to L\to \Gamma\to G\to 1)
with L a free abelian group equipped with a G‑action, the LHS spectral sequence takes the form
(E_2^{p,q}=H^p(G, H^q(L,\mathbb Z))\Rightarrow H^{p+q}(\Gamma,\mathbb Z)).
If all differentials (d_r^{p,q}) up to a certain page r − 1 vanish, the first potentially non‑zero differential (d_r^{p,q}) is obstructed by a characteristic class
(v_r^q\in H^r(G, H^{q-r+1}(L,\mathbb Z))).
These classes encode precisely the failure of the spectral sequence to collapse at the earlier page.

The central contribution of the work is to relate the characteristic classes of a lattice that decomposes as a direct sum of G‑invariant sublattices,
(L=\bigoplus_{i=1}^k L_i),
to the characteristic classes of the individual summands. Using the naturality of the LHS spectral sequence together with the Künneth theorem for exterior algebras, the authors show that the cohomology algebra (H^*(L,\mathbb Z)) splits as a tensor product of the cohomologies of the summands. Consequently, the differentials in the spectral sequence respect this tensor product structure: each differential is the sum of the differentials acting on the individual factors.

The main theorem (Theorem 3.1) states that, provided the differentials up to page r − 1 are zero, the characteristic class on the whole lattice is the sum of the induced classes from the summands: \