Bousfield localization functors and Hopkins’ chromatic splitting conjecture.

*(English)*Zbl 0830.55004
Cenkl, Mila (ed.) et al., The Čech centennial. A conference on homotopy theory dedicated to Eduard Čech on the occasion of his 100th birthday, June 22-26, 1993, Northeastern University, Boston, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 181, 225-250 (1995).

The content is well described by the following extracts from the introduction.

“The major result of this paper is that finite torsion spectra are local with respect to any infinite wedge of Morava \(K\)-theories \(\bigvee_{i< \infty} K(n_i)\). This has several interesting corollaries. For example, it implies that there are no maps from the Johnson-Wilson spectra \(\text{BP} \langle n\rangle\) to a finite spectrum. It also implies that if \(E\) is a ring spectrum which detects all finite spectra, so that \(E_* (X)\neq 0\) if \(X\) finite, then \(L_E X\) is either \(X\) or \(X_p\), the \(p\)-completion of \(X\), for finite \(X\). This in turn implies that the only smashing localization which detects all finite complexes is the identity functor.”

“The last section of the paper discusses the consequences of the chromatic splitting conjecture on the homotopy groups of \(L_n S^0\). We show that, given the chromatic splitting conjecture, the divisible summands in \(\pi_* L_n S^0\) for \(n\geq 1\) can be determined. There are \(3^{n-1}\) of them, with \(2^{n-1}\) of them occuring in dimension \(-2n\), and the others spread out from dimension \(-2n-1\) to dimension \(-n^2-1\). This therefore explains part of the Shimomura-Yabe calculation of \(\pi_* L_2 S^0\) for \(p>3\)”.

For the entire collection see [Zbl 0809.00023].

“The major result of this paper is that finite torsion spectra are local with respect to any infinite wedge of Morava \(K\)-theories \(\bigvee_{i< \infty} K(n_i)\). This has several interesting corollaries. For example, it implies that there are no maps from the Johnson-Wilson spectra \(\text{BP} \langle n\rangle\) to a finite spectrum. It also implies that if \(E\) is a ring spectrum which detects all finite spectra, so that \(E_* (X)\neq 0\) if \(X\) finite, then \(L_E X\) is either \(X\) or \(X_p\), the \(p\)-completion of \(X\), for finite \(X\). This in turn implies that the only smashing localization which detects all finite complexes is the identity functor.”

“The last section of the paper discusses the consequences of the chromatic splitting conjecture on the homotopy groups of \(L_n S^0\). We show that, given the chromatic splitting conjecture, the divisible summands in \(\pi_* L_n S^0\) for \(n\geq 1\) can be determined. There are \(3^{n-1}\) of them, with \(2^{n-1}\) of them occuring in dimension \(-2n\), and the others spread out from dimension \(-2n-1\) to dimension \(-n^2-1\). This therefore explains part of the Shimomura-Yabe calculation of \(\pi_* L_2 S^0\) for \(p>3\)”.

For the entire collection see [Zbl 0809.00023].

Reviewer: C.A.Robinson (Coventry)

##### MSC:

55P42 | Stable homotopy theory, spectra |

55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |

55P60 | Localization and completion in homotopy theory |

55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |