The equivariant Hurewicz map

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a compact Lie group, <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> be a based <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-space, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-representation. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript upper V Superscript upper G Baseline left-parenthesis upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>π<!-- π --></mml:mi> <mml:mi>V</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi _V^G(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the equivariant homotopy group of <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> in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript upper V Superscript upper G Baseline left-parenthesis upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>V</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H_V^G(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the equivariant ordinary homology group of <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> with Burnside ring coefficients in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then there is an equivariant Hurewicz map <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h colon pi Subscript upper V Superscript upper G Baseline left-parenthesis upper Y right-parenthesis right-arrow upper H Subscript upper V Superscript upper G Baseline left-parenthesis upper Y right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>:</mml:mo> <mml:msubsup> <mml:mi>π<!-- π --></mml:mi> <mml:mi>V</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>V</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h:\pi _V^G(Y) \to H_V^G(Y).</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> One should not expect this map to be an isomorphism, since <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript upper V Superscript upper G Baseline left-parenthesis upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>V</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H_V^G(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must be a module over the Burnside ring, but <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript upper V Superscript upper G Baseline left-parenthesis upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>π<!-- π --></mml:mi> <mml:mi>V</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi _V^G(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> need not be. However, here it is shown that, under the obvious connectivity conditions on <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>, this map induces an isomorphism between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript upper V Superscript upper G Baseline left-parenthesis upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>V</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H_V^G(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an algebraically defined modification of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript upper V Superscript upper G Baseline left-parenthesis upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>π<!-- π --></mml:mi> <mml:mi>V</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi _V^G(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The equivariant Freudenthal Suspension Theorem contains a technical hypothesis that has no nonequivariant analog. Our results shed some light on the behavior of the suspension map when this rather undesirable technical hypothesis is not satisfied.

Locations

• Transactions of the American Mathematical Society - View - PDF