Loop space decompositions of $$(2n-2)$$-connected $$(4n-1)$$-dimensional Poincaré Duality complexes
Ruizhi Huang, Stephen Theriault
Abstract
Abstract Beben and Wu showed that if M is a $$(2n-2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> -connected $$(4n-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mn>4</mml:mn><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> -dimensional Poincaré Duality complex such that $$n\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math> and $$H^{2n}(M;{{\mathbb {Z}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>M</mml:mi><mml:mo>;</mml:mo><mml:mi>Z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> consists only of odd torsion, then $$\Omega M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Ω</mml:mi><mml:mi>M</mml:mi></mml:mrow></mml:math> can be decomposed up to homotopy as a product of simpler, well-studied spaces. We use a result from Beben and Theriault (Doc Math 27:183-211, 2022) to greatly simplify and enhance Beben and Wu’s work and to extend it in various directions.