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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

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Information about Milnor conjecture

According to David Roberts comment and the following paper it is open for dimensions $n\geq 4$. Pan, Jiayin, A proof of Milnor conjecture in dimension 3, J. Reine Angew. Math. 758, 253-260 (2020). ZBL …
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De Rham decomposition theorem, generalisations and good references

Response to the first question: Pantilie, Radu, A simple proof of the de Rham decomposition theorem, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 36, No. 3-4, 341-343 (1992). ZBL0811.53040.
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