date: 2022-02-23 18:45 modification date: Friday 1st April 2022 21:26:14 title: “Hurewicz” aliases: [Hurewicz theorem, generalized Eilenberg-Maclane spectrum, generalized Eilenberg-Maclane spectra]

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  • #homotopy
• Refs:
  • #todo/add-references
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  • Adams resolution

Hurewicz

Given a space \(X\), define a family of maps \begin{align*} h_k: \pi_k X \to H_k X \\ [f] \mapsto f_*(\mu_k) \end{align*} where \(H_k X = \langle \mu_k \rangle\).

If \(X\) is \(n-1\) connected where \(n\geq 2\), then \(h_k\) is an isomorphism for all \(k \leq n\).

In particular, \(\pi_n X \cong H_n X\) as groups.

Proof using spectral sequences: https://people.math.wisc.edu/~maxim/spseq.pdf#page=5

Necessity of simple-connectivity assumption: see the Poincare homology sphere

Relation to the Whitehead theorem:

For spectra