nLab Steenrod representability

Idea

Given a topological space $X$, a cycle in the ordinary (or generalized) homology of $X$ is called Steenrod representable [Landweber 1967] if it is the image of the fundamental class of a manifold $\Sigma$ under pushforward along a continuous map $\phi \colon \Sigma \longrightarrow X$.

In physics (string theory), if one thinks of $X$ as (“target”) spacetime, of $\Sigma$ as (a factor space of) the abstract worldvolume of a brane and of $\phi \colon \Sigma \longrightarrow X$ as the actual trajectory of the brane in spacetime, then this situation is referred to as the brane “wrapping” the cycle.

References

• Peter S. Landweber: Steenrod representability of stable homology, Proc. Amer. Math. Soc. 18 (1967) 523-529 [doi:10.1090/S0002-9939-1967-0229241-4, doi:10.2307/2035491, pdf]

See also:

• Andrés Angel, Arley Fernando Torres, Carlos Segovia, section 8 of: $\mathbb{Z}_k$-stratifolds, Algebr. Geom. Topol. 24 (2024) 1863-1901 [arXiv:1810.00531, doi:10.2140/agt.2024.24.1863]

• Diarmuid Crowley, Mark Grant: Immersed but not embedded homology classes [arXiv:2412.15359]