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Atiyah-Hirzebruch spectral sequence

{title2=$E^2_{p,q}=\widetilde H_p(X;\pi_qE)$}

The skeletal filtration of a <CW complex> and the <cofiber sequences of spectra> of its cell attachments produce
$$
E^2_{p,q}=\widetilde H_p(X;\pi_qE)\Longrightarrow\widetilde E_{p+q}(X),\qquad d_r:E^r_{p,q}\to E^r_{p-r,q+r-1}.
$$
For a <connective spectrum> it is first quadrant and converges in each fixed total degree. Its reduced version has no contribution from the basepoint summand.

Homological Atiyah-Hirzebruch spectral sequence

Bottom-degree generalized Hurewicz isomorphism

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 22 / 4 / a / Solution

Homological Atiyah-Hirzebruch spectral sequence ( $E_{p, q}^{2} = H_{p} (X; π_{q} E)$ )

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Homological Atiyah-Hirzebruch spectral sequence (Ep,q2​=Hp​(X;πq​E))

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Homological Atiyah-Hirzebruch spectral sequence ( $E_{p, q}^{2} = H_{p} (X; π_{q} E)$ )

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