Cobordism of nested manifolds

We study cobordisms of nested manifolds, which are manifolds together with embedded submanifolds, which can themselves have embedded submanifolds, etc. We identify a nested analog of the Pontryagin-Thom construction. Moreover, when the highest-dimensional manifold has a normal bundle with a framed direction, we find spaces homotopy equivalent to the nested Pontryagin-Thom spaces that relate nested manifolds up to cobordism with links up to cobordism. This gives rise to nested cobordism invariants coming from previously studied cobordism invariants of links. In addition, we provide an alternative proof of a result by Wall about the splitting of the stable nested cobordism groups.

💡 Research Summary

The paper develops a cobordism theory for “nested manifolds,” i.e., a manifold together with an embedded submanifold, which itself may contain further embedded submanifolds, and so on. After reviewing the classical Pontryagin–Thom construction, the author defines a once‑nested (θ′, θ)‑submanifold K′⊂K⊂M, where K is a θ‑submanifold of a closed m‑manifold M and K′ is a θ′‑submanifold of K. A nested cobordism consists of a cobordism W between the outer manifolds together with a sub‑cobordism W′⊂W between the inner ones, each equipped with lifts of their normal bundle classifying maps to the appropriate structure spaces B and B′.

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