Loop space decompositions of $$(2n-2)$$-connected $$(4n-1)$$-dimensional Poincaré Duality complexes

Abstract

Beben and Wu showed that if M is a $$(2n-2)$$ ( 2 n - 2 ) -connected $$(4n-1)$$ ( 4 n - 1 ) -dimensional Poincaré Duality complex such that $$n\ge 3$$ n ≥ 3 and $$H^{2n}(M;{{\mathbb {Z}}})$$ H 2 n ( M ; Z ) consists only of odd torsion, then $$\Omega M$$ Ω M 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.

Loop homotopy of 6–manifolds over 4–manifolds

“…When k is odd, the loop decomposition of the Poincaré complex P 4 .k/ [ e 7 was determined by Huang and Theriault [20]. For any prime p, let S m fp r g be the homotopy fibre of the degree p r map on S m .…”

“…From the suspension viewpoint, So and Theriault [31] determined the homotopy type of the suspension of connected 4-manifolds, while Huang [19] studied the suspension of simply connected 6-manifolds. From the loop viewpoint, Beben and Theriault [6] studied the loop decompositions of .n 1/-connected 2n-manifolds, while Beben and Wu [8] and Huang and Theriault [20] studied the loop decompositions of the .n 1/-connected .2nC1/-manifolds. The homotopy groups of these manifolds were also investigated by Samik Basu and Somnath Basu [3; 4] from different point of view.…”

“…Further note that Theorem 1.4 is only a partial result. The difficulty in this case is due to the fact that the proof of Theorem 1.4 heavily relies on a result of Huang and Theriault [20] on the loop decomposition of 2-connected 7-manifolds. As discussed in [20, Section 6], the case when k D 2 r m with m odd and greater than 1 is much more difficult.…”

Suspension homotopy of 6–manifolds

“…There have been several interesting investigations recently in this direction. For instance, Beben and Theriault [6] studied the loop decompositions of .s 1/-connected 2s-manifolds, while Beben and Wu [8] and Huang and Theriault [17] studied the loop decompositions of the .s 1/-connected .2sC1/-manifolds. The homotopy groups of these manifolds were also investigated by Samik Basu and Somnath Basu [2; 3] from a different point of view.…”