Atiyah–Hirzebruch Spectral Sequence (AHSS)

• AHSS is a spectral sequence that computes generalized (co)homology of cell-filtered spaces using ordinary (co)homology with twisted coefficients.
• It employs both homological and cohomological constructions, with differentials reflecting cellular boundaries and incorporating Spanier–Whitehead duality.
• The method underpins classical Poincaré duality and aids in analyzing torsion and extension problems in generalized (co)homology theories.

The Atiyah–Hirzebruch spectral sequence (AHSS) is a spectral sequence that computes generalized homology or cohomology of spaces with a cell filtration, expressing these (co)homology groups in terms of their ordinary (co)homology with coefficients in the homotopy groups of the spectrum representing the generalized (co)homology theory. Spanier–Whitehead duality interrelates the homological and cohomological versions of the AHSS, and this duality underpins Poincaré duality at the level of spectral sequences for oriented ring spectra—recovering classical parallels between homology and cohomology in the context of generalized theories.

1. Homological Atiyah–Hirzebruch Spectral Sequence

Given a finite CW-spectrum $X$ filtered by skeleta

$$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$

and a ring spectrum $\cE$, the homological AHSS is produced by applying $\cE$-homology to the cofiber sequences $X^{(p-1)} \to X^{(p)} \to X^{(p)}/X^{(p-1)}$. Each cofiber is a wedge of spheres, yielding

$E^1_{p,q} = \cE_{p+q}(X^{(p)}/X^{(p-1)}) \cong \bigoplus_{I_p} \cE_q(pt),$

where $I_p$ indexes the $p$-cells. The differential $d_1$ is the cellular boundary, so that

$E^2_{p,q} = H_p(X; \cE_q(pt)).$

The spectral sequence converges to the graded pieces of $$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$0, and higher differentials $$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$1 have bidegree $$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$2, i.e.,

$$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$3

This approach expresses generalized homology as the abutment of a spectral sequence with $$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$4 page the ordinary homology with twisted coefficients.

2. Cohomological Atiyah–Hirzebruch Spectral Sequence

The cohomological AHSS is derived from applying cohomology to the pairs $$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$5, leading to an analogous construction: $$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$6 and subsequent differentials in the exact couple produce

$$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$7

with higher $$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$8 of bidegree $$\varnothing = X^{(-1)} \subset X^{(0)} \subset \cdots \subset X^{(n)} = X,$$9: $\cE$0 This structure converges to $\cE$1, the generalized cohomology of $\cE$2.

3. Spanier–Whitehead Duality and AHSS Isomorphism

For finite spectra, the Spanier–Whitehead duality functor $\cE$3 establishes an equivalence

$\cE$4

The main duality theorem states that this isomorphism extends to all pages $\cE$5 of the AHSS,

$\cE$6

compatible with differentials: $\cE$7 On the $\cE$8 page, this is induced by the coefficient isomorphisms, explicitly relating ordinary cohomology on $\cE$9 to ordinary homology on $\cE$0.

4. Sketch of Proof Structure

The duality between the homological and cohomological AHSS is established by constructing a natural transformation between the exact couples governing both spectral sequences, with the Spanier–Whitehead duality isomorphism intertwining their structures. At the level of the quotient terms $\cE$1 and $\cE$2, one identifies

$\cE$3

and verifies naturality of all structure maps. The standard spectral sequence comparison lemma ensures that such an isomorphism of exact couples induces isomorphisms $\cE$4 at every page.

5. Application: Poincaré Duality in Generalized (Co)homology

If $\cE$5 is a finite $\cE$6-dimensional Poincaré duality complex oriented over $\cE$7, the Spivak normal fibration admits a Thom–Dold class, and the Thom isomorphism

$\cE$8

can be followed by Spanier–Whitehead duality to yield

$\cE$9

This is precisely Poincaré duality for $X^{(p-1)} \to X^{(p)} \to X^{(p)}/X^{(p-1)}$0-homology and cohomology,

$X^{(p-1)} \to X^{(p)} \to X^{(p)}/X^{(p-1)}$1

and this identification holds at every page of the AHSS via the isomorphism

$X^{(p-1)} \to X^{(p)} \to X^{(p)}/X^{(p-1)}$2

Both spectral sequences are thus naturally identified, and their abutments compute the same graded group up to the Poincaré duality shift.

6. Consequences and Related Results

• The duality holds for any finite CW spectrum and any spectrum $X^{(p-1)} \to X^{(p)} \to X^{(p)}/X^{(p-1)}$3.
• For classical (co)homology ($X^{(p-1)} \to X^{(p)} \to X^{(p)}/X^{(p-1)}$4), this recovers ordinary Poincaré duality.
• The construction operates at the level of spectral sequences, providing a highly structured approach to duality phenomena in algebraic topology.
• The compatibility ensures that computations done in homology or cohomology are transferable via duality, facilitating the analysis of torsion and extension problems in generalized (co)homology.

The main results and constructions summarized above are developed in "A note on Duality and the Atiyah–Hirzebruch spectral sequence" (Hans, 24 Jul 2025). The general theory and connections to Poincaré duality and Spanier–Whitehead duality as it interacts with the AHSS are expounded in detail within that paper.