Cohomotopy sets of 4-manifolds. (English) Zbl 1273.57012

Kirby, Robion; Melvin, Paul; Teichner, Peter

Kirby, Rob (ed.) et al., Proceedings of the Freedman Fest. Based on the conference on low-dimensional manifolds and high-dimensional categories, Berkeley, CA, USA, June 6–10, 2011 and the Freedman symposium, Santa Barbara, CA, USA, April 15–17, 2011 dedicated to Mike Freedman on the occasion of his 60th birthday. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 18, 161-190 (2012).

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The purpose of this paper is to classify the cohomotopy set \(\pi ^{n}(X)\) of the free homotopy classes of maps from a 4-manifold \(X\) to the \(n\)-sphere \(S^{n}\). For a smooth, closed, connected and oriented 4-manifold the Pontryagin-Thom construction induces a refinement of Poincaré duality, i.e. there exists the commutative diagram \[ \begin{tikzcd} \pi^{n}(X)\rar["{\cong}"]\dar["h^n" '] & \mathbb{F}_{4-n} (X)\dar["h_{4-n}"] \\ H^n(X) \rar["{\cong}" '] & H_{4-n} (X) \rlap{\,,} \end{tikzcd} \] where \(F_{k}(X)\) is the set of closed \(k\)-dimensional submanifolds of \(X\) with a framing on their normal bundle, up to normally framed bordism in \(X\times [0,1]\).
Using the simple geometric representations, the authors obtain interesting results which describe the group structure of the homotopy classes of maps from \(X\) to the 3-sphere, enumerate the homotopy classes of maps from \(X\) to the 2-sphere and prove that \(h^{n}\) is an isomorphism in all other cases.
For the entire collection see [Zbl 1253.00022].

Reviewer: Vladimir Baladze (Batumi)

MSC:

57N13

Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)

Keywords:

4-manifolds; cohomotopy; framed submanifolds