The transfer and compact Lie groups

Abstract

Let <italic>G</italic> be a compact Lie group with <italic>H</italic> and <italic>K</italic> arbitrary closed subgroups. Let <italic>BG, BH, BK</italic> be <italic>l</italic>-universal classifying spaces, with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho left-parenthesis upper H comma upper G right-parenthesis colon upper B upper H right-arrow upper B upper G"> <mml:semantics> <mml:mrow> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>B</mml:mi> <mml:mi>H</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>B</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\rho (H,G):BH \to BG</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the natural projection. Then transfer homomorphisms <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis upper H comma upper G right-parenthesis colon h left-parenthesis upper B upper H right-parenthesis right-arrow h left-parenthesis upper B upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>B</mml:mi> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>B</mml:mi> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T(H,G):h(BH) \to h(BG)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are defined for <italic>h</italic> an arbitrary cohomology theory. One of the basic properties of the transfer for finite coverings is a double coset formula. This paper proves a double coset theorem in the above more general context, expressing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Superscript asterisk Baseline left-parenthesis upper K comma upper G right-parenthesis ring upper T left-parenthesis upper H comma upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∘<!-- ∘ --></mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\rho ^{\ast }}(K,G) \circ T(H,G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a sum of other compositions. The main theorems were announced in the Bulletin of the American Mathematical Society in May 1977.

Locations

• Project Euclid (Cornell University) - View - PDF
• Transactions of the American Mathematical Society - View - PDF