nLab
phantom map

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Contents

Definition

In stable homotopy theory/cohomology theory a homomorphism of spectra EFE \longrightarrow F is called a phantom map if it induces (up to homotopy) the zero morphism on generalized homology theories E F E_\bullet \longrightarrow F_\bullet, hence if for every topological space/homotopy type XX the induced map of homology groups

E (X)F (X) E_\bullet(X) \longrightarrow F_\bullet(X)

is the zero morphism.

The existence of phantom maps implies that despite the Brown representability theorem, there is a subtle difference between generalized (Eilenberg-Steenrod) homology theories and the spectra which represent them: the latter contain in general more information. (Note that this issue is not present when working with cohomology theories instead of homology theories).

In even more classical context, a continuous map ff from a CW-complex XX to a topological space YY is a phantom map if it is not homotopic to a constant but for every finite CW-subcomplex ZXZ\subset X the restriction f| Z:ZYf|_Z:Z\to Y is homotopic to a constant.

Properties

Between Landweber-exact spectra

Between Landweber exact spectra, every phantom map is already null-homotopic. (e.g. Lurie, 10, corollary 7).

References

Last revised on April 4, 2017 at 15:52:33. See the history of this page for a list of all contributions to it.