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Complex shifts and spectral homotopy indices

PDF p. 4: Shift [2j] in equation (2.1)

Statement

Let $C$ be an unbounded cochain complex of abelian groups. Write $H C$ for the spectrum associated, through the stable spectra interface, to the Eilenberg–Mac Lane Omega spectrum of the chain complex $D_{n} = C^{- n}$ . Then, naturally in $C$ and for all integers $i, j$ ,

$π_{i} (H C) ≅ H^{- i} (C), π_{i} (H (C [2 j])) ≅ H^{2 j - i} (C), π_{i} ((H C) [2 j]) ≅ H^{2 j - i} (C) .$

Here $C [2 j]$ is the cochain shift, whereas $(H C) [2 j]$ is spectral suspension by $2 j$ ; the formulas do not identify these two objects by a specified equivalence. Consequently, a given equivalence ${gr}^{j} E ≃ (H C_{j}) [2 j]$ induces $π_{i} {gr}^{j} E ≅ H^{2 j - i} (C_{j})$ . The unshifted realization of a support complex $C_{j, T}$ has $π_{i} = H^{- i} (C_{j, T})$ ; its $2 j$ -shift has the latter shifted index.

Facts & Assumptions

Given: $C$ , $i$ , and $j$ as stated; for the graded conclusion, a specified equivalence ${gr}^{j} E ≃ (H C_{j}) [2 j]$ .

Goal: Establish the displayed natural group identifications and their graded and support consequences.

[F1]

Reindexing cochains as chains identifies $H_{i} (D)$ with $H^{- i} (C)$ .

[F2]

Eilenberg–Mac Lane spectrum of a chain complex supplies the simplicial Omega spectrum of $D$ and its stable homotopy groups.

[F3]

Reading spectrum homotopy from an Omega tower compares a coherent Omega tower with its associated spectrum.

[F4]

Cohomology of an integer shift computes cohomology after cochain shift, and Reindexing commutes with shifts identifies the corresponding shifted chain complex.

[F5]

Homotopy groups of integer spectral shifts computes homotopy groups after spectral shift; Stable spectra and exact triangles supplies the associated stable category and homotopy-group functors.

Proof

1.1F1F2F3F5given

Apply [F2] to $D_{n} = C^{- n}$ . Its simplicial Omega spectrum defines a coherent Omega tower of pointed spaces and hence, by the stable spectra interface, an associated spectrum $H C$ . By [F3], its enhanced homotopy groups agree with the stable groups in [F2]; by [F1] these are $H^{- i} (C)$ . These identifications are natural in maps of complexes.

2.1F4F5givenstep 1.1∎

Apply step 1.1 to $C [2 j]$ . By [F4], its associated chain complex is the corresponding chain shift and $H^{- i} (C [2 j]) ≅ H^{2 j - i} (C)$ . Independently, [F5] gives $π_{i} ((H C) [2 j]) ≅ π_{i - 2 j} (H C) ≅ H^{2 j - i} (C)$ . A specified equivalence of the graded object induces an isomorphism on homotopy groups, proving the graded formula. Applying the same two formulas to the support complex proves the final assertion, with its unshifted and shifted spectra kept distinct.

Depends on

Used by · 6

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