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General framework to compute Wu invariants for immersions of 3-manifolds into 5-space

Develop a general methodological framework to compute the Wu invariant c_τ(f) for immersions f: M^3 → R^5 of oriented 3-manifolds, addressing its dependence on the choice of parallelization and providing systematic procedures for selecting or handling parallelizations so that c_τ(f) becomes computable.

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Background

The classification of immersions M3 → R5 up to regular homotopy uses two invariants: the Wu invariant and a Smale-type invariant. While Saeki–Szűcs–Takase provided geometric formulas making the Smale-type invariant computable, the Wu invariant has remained difficult to compute.

The authors highlight that a general framework for computing the Wu invariant is lacking, principally because c_τ(f) depends on a chosen parallelization τ of M3 and can take all possible values when τ varies, making the choice of parallelization crucial.

In this paper, the authors introduce almost contact parallelizations and prove vanishing results for special classes (e.g., contact embeddings and certain complex-tangency situations), and they analyze switching across parallelizations in the appendix. Nonetheless, a general computational framework remains to be established.

References

In contrast, a framework to compute the Wu invariant has not yet been established. It seems to be due to the following reason: the Wu invariant is defined for a fixed parallelization (trivialization of the tangent bundle) of the 3-manifold. Furthermore, the Wu invariant takes all possible values by changing parallelizations (Appendix \ref{app:switch}). Hence, it is crucial to choose a nice parallelization depending on situation.

— Regular homotopy classes of links of simple singularities and immersions associated with their Dynkin diagrams (2405.02513 - Tanabe, 3 May 2024) in Section 1 (Introduction)