Local structure of homogeneous 𝐴𝑁𝑅-spaces

We investigate to what extend finite-dimensional homogeneous locally compact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper N upper R"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>N</mml:mi> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">ANR</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spaces have common properties with topological manifolds. Specially, the local structure of homogeneous <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper N upper R"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>N</mml:mi> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">ANR</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spaces is described. Using that description, we provide a positive solution of the problem whether every finite-dimensional homogeneous metric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper N upper R"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>N</mml:mi> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">ANR</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -compactum <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is dimensionally full-valued, i.e. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension upper X times upper Y equals dimension upper X plus dimension upper Y"> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>X</mml:mi> <mml:mo> × </mml:mo> <mml:mi>Y</mml:mi> <mml:mo>=</mml:mo> <mml:mi>dim</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>X</mml:mi> <mml:mo>+</mml:mo> <mml:mi>dim</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\dim X\times Y=\dim X+\dim Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any metric compactum <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .