Parametrised loop coproduct obstruction conjecture

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Let E→BE\to B be a smooth fiber bundle with fiber a smooth closed manifold MM, and let f:E→M×Bf:E\to M\times B be a fiberwise homotopy equivalence over BB. Suppose there are spectral operations in families, defined as morphisms of parametrised spectra, including Δfib⁡E\Delta_{\operatorname{fib}}^E, Δfib⁡B×M\Delta_{\operatorname{fib}}^{B\times M}, ΞrB\Xi_{{{r}}}^B, and ΞlB\Xi_{{{l}}}^B. Parametrised loop coproduct obstruction conjecture. An analogue of the corresponding loop coproduct formula holds:

Δfib⁡B×M∘f!−f∧f∘Δfib⁡E=ΞrB−ΞlB.\Delta_{\operatorname{fib}}^{B\times M} \circ f_! - f\wedge f \circ \Delta_{\operatorname{fib}}^E = \Xi_{{{r}}}^B - \Xi_{{{l}}}^B.

This conjecture extends the obstruction formula from a single manifold to smooth fiber bundles and parametrised spectra. The source gives no evidence that it has been proved or disproved.

References

Primary source

Lea Kenigsberg and Noah Porcelli, “Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory”, arXiv:2407.13662 (2026).

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