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In this note we will prove that, given an      R ∞  ( Q ∞  ) {R^\infty }({Q^\infty })      -manifold      M M      , there is a deformation of      Homo ⁡  ( M ) \operatorname {Homo}(M)       into      Homeo ⁡  ( M ) \operatorname {Homeo}(M)       whose final stage is a weak homotopy equivalence, and that if      M M       has the homotopy type of a finite simplicial complex, then      Homeo ⁡  ( M ) \operatorname {Homeo}(M)       is an      ANE ⁡  ( C W ( M ) ) \operatorname {ANE}(\mathcal {C}\mathcal {W}(\mathcal {M}))       and an      ANE ⁡  ( C W ( C ) , G δ  ) \operatorname {ANE}(\mathcal {C}\mathcal {W}(\mathcal {C}),{G_\delta })      .

Some homotopy properties of the homeomorphism groups of 𝑅^{∞} (𝑄^{∞})-manifolds