nLab stable splitting

Idea

In algebraic topology, by stable splitting one refers to a situation where (the homotopy type of) a topological space or (the stable homotopy type of) a spectrum $X$ becomes weakly homotopy equivalent to a wedge sum of (generally simpler) spaces/types after some number $n \in \mathbb{N} \sqcup \{\infty\}$ of (reduced) suspensions $\Sigma(-)$:

$$\Sigma^n X \;\simeq\; \bigvee_i Y_i \mathrlap{\,.}$$

Examples

Basic examples

Example

(stable splitting of product spaces)
For $X$ and $Y$ a pair of pointed CW-complexes, the reduced suspension of their product space is homotopy equivalent to the wedge sum of their individual reduced suspensions with that of their smash product $X \wedge Y$:

$$\Sigma(X \times Y) \,\underset{hmtp}{\simeq}\, \Sigma X \,\vee\, \Sigma Y \,\vee\, \Sigma(X \wedge Y) \mathrlap{\,.}$$

(cf. Hatcher 2002 Prop. 4I.1)

Of mapping spaces

Cf. stable splitting of mapping spaces

Of Manifolds

On stable splitting for 6-manifolds (such as Calabi-Yau 3-folds) see Huang 2023.

References

Textbook account:

• Allen Hatcher, §4.I in: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]

On stable splitting of 6-manifolds:

• Ruizhi Huang: Suspension homotopy of 6-manifolds, Algebr. Geom. Topol. 23 (2023) 439-460 [arXiv:2104.04994, doi:10.2140/agt.2023.23.439]