Extending intersection homology type invariants to non-Witt spaces

Intersection homology with field coefficients: K‐Witt spaces and K‐Witt bordism

“…Banagl [2] has since extended duality even further by identifying conditions on which non-Witt spaces possess self-duality (conditions equivalent to the existence of certain towers of Lagrangian structures on strata), but Witt spaces remain an important class of examples defined by a relatively tractable condition.…”

“…In this case, the Atiyah-Hirzebruch E 2 terms look like H r (X; Ω Zp−Witt s ⊗ Z (2) ). Since Ω Zp−Witt s is 2-torsion for s > 0, in this case the terms are identical to those from the spectral sequence before the localization, while the (r, 0) term becomes H r (X; Z (2) ).…”

“…In this case, the Atiyah-Hirzebruch E 2 terms look like H r (X; Ω Zp−Witt s ⊗ Z (2) ). Since Ω Zp−Witt s is 2-torsion for s > 0, in this case the terms are identical to those from the spectral sequence before the localization, while the (r, 0) term becomes H r (X; Z (2) ). Now, this spectral sequence degenerates at the E 2 term by a theorem of Taylor and Williams [29,Section 2] that states that any 2-local MSO-module spectrum (in this case (W p ) (2) ) splits as a wedge of Eilenberg-MacLane spectra.…”

“…The group Ω Zp−Witt n (X) must be finitely generated (since everything else in the spectral sequence is), so, by basic facts about localization, we can read off the 2-primary and infinite cyclic summands of Ω Zp−Witt (2) . In particular, any 2-primary components are shared between the two, and Z summands of the former, correspond bijectively with Z (2) terms in the latter, which can come only from the summand H n (X; Z (2) ).…”

“…In particular, any 2-primary components are shared between the two, and Z summands of the former, correspond bijectively with Z (2) terms in the latter, which can come only from the summand H n (X; Z (2) ). Thus, modulo odd torsion, we must have Ω Zp−Witt…”