The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture

Abstract

An IP-space is a pseudomanifold whose defining local properties imply that its middle perversity global intersection homology groups satisfy Poincaré duality integrally. We show that the symmetric signature induces a map of Quinn spectra from IP bordism to the symmetric L-spectrum of Z, which is, up to weak equivalence, an E∞ ring map. Using this map, we construct a fundamental L-homology class for IP-spaces, and as a consequence we prove the stratified Novikov conjecture for IP-spaces whose fundamental group satisfies the Novikov conjecture.2010 Mathematics Subject Classification. 55N33, 57R67, 57R20, 57N80, 19G24.

$L$-homology on ball complexes and products

“…The first formula is also proved in [BLM14,Theorem 11.1] for IP-spaces. A special case of the second formula gives that for a closed k-dimensional manifold F and a closed n-dimensional manifold X we have a commutative diagram N (X)…”

“…The present paper differs from [LM14,BLM14] in that although we work with ball complexes we do not use the ad-theories. We work instead in the setup of additive categories with chain duality from [Ran92] and [Wei92].…”

“…As a byproduct they constructed a cohomology theory on the category of ball complexes X → H n (X; L • ) ∼ = L n (Λ * (X)), where Λ * (X) is a certain algebraic bordism category (different from Λ * (X)), see Theorem 16.1 and Remark 16.2 in [LM14]. Moreover, in the preprint [BLM14] Banagl, Laures and McClure use the ad-theories to construct a homology theory Z → H n (Z; L • ) on the category of topological spaces with the coefficients in the symmetric L-theory spectrum L • and prove a product formula for symmetric signatures of IP-spaces. However, they do not construct a homology theory with the coefficients in the quadratic spectrum L • and they also do not consider the quadratic signature over X map (1.1).…”

On the homotopy type of L‐spectra of the integers

“…The expectation that the isomorphisms are induced by a map of spectra was finally implemented by Banagl, Laures and McClure [3], informally by sending an IP‐space to its intersection cochains of middle perversity.…”

Topological and Hodge L-classes of singular covering spaces and varieties with trivial canonical class

“…In fact, this phenomenon has been used by Wall to construct Poincaré spaces that are not homotopy equivalent to manifolds. In [50], he constructs 4dimensional Poincaré CW-complexes X with cyclic fundamental group of prime order such that the signature of both X and its universal cover is 8. Such examples show that multiplicativity of the signature under finite covers is a subtle matter in the presence of singularities.…”