A-PHANTOM MAPS

Abstract. We investigate what happens if one replaces S^{0} by an arbitrary space A in the definition of phantom maps.

0.1. Definition. A map f: X → Y is an A-phantom map if the composition

X ⟶^{f} Y → P_{Σ^{n}A}Y

is null homotopic for all n. Let P h_{A}(X, Y) ⊂ [X, Y] denote the set of homotopy classes of phantom maps from X to Y.

Equivalently, P_{Σ^{n}A}(f) ≃ ∗ for all n?

0.2. Example. Any map into a finite complex is BZ/p-phantom; e.g. all maps CP^{∞} → S^{3}.

0.3. Proposition. Any A-phantom map is an S^{0}-phantom map (i.e. a phantom map in the usual sense).

Just as in the usual case P h_{A}(X, Y) = lim_{n}^{1}[X, ΩP_{Σ^{n}A}Y].

Here are some problems:

• Find conditions on A and B that ensure A-phantom implies B-phantom. Find a B-phantom map that is not an A-phantom map.

• Find an example where the subsets of A-phantom maps for various spaces A determine an interesting structure in the set P h(X, Y) of all phantom maps from A to B.

• Say that two spaces have the same A-n type for all n if P_{Σ^{n}A}X ≃ P_{Σ^{n}A}Y for all n. Will this give interesting subsets of SNT (X)?

• Say that a map f: X → X is a weak A-identity if P_{Σ^{n}A}(f) is homotopic to the identity on P_{Σ^{n}A}(X) for all n. Study the group W I_{A}(X) of weak A-identities. If X is an Hspace and f a weak A-identity, then presumably 1 + f is A-phantom.

When searching for examples of phantoms it is helpful to keep this in mind:

0.4. Proposition. If H^{j}(X; π_{j+1}(Y) ⊗ Q) = 0 for all j ≥ 0 then P h(X, Y) = ∗.

Proof. The assumption implies that [X, ΩP_{S^{n}}Y] is finite group for all n. By Mittag-Leffler P h(X, Y) = lim^{1}[X, ΩP_{S^{n}}Y] = ∗.

References

[1] C.A. McGibbon, Phantom maps, Handbook of algebraic topology (1995), 1209–1257, North-Holland.

[2] C.A. McGibbon and J.M. Møller, A note on connected covers and Neisendorfer’s localization theorem, Fund. Math. 152 (1997), 211–230.