nLab phantom map

Definition

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

$$E_\bullet(X) \longrightarrow F_\bullet(X)$$

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 $f$ from a CW-complex $X$ to a topological space $Y$ is a phantom map if it is not homotopic to a constant but for every finite CW-subcomplex $Z\subset X$ the restriction $f|_Z:Z\to Y$ 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

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 17 Phantom maps (pdf)