We study the topological $4$-dimensional surgery problem for a closed connected orientable topological $4$-manifold $X$ with vanishing second homotopy and $\pi_1(X)\cong A * F(r)$, where $A$ has one end and $F(r)$ is the free group of rank $r\ge 1$. Our result is related to a theorem of Krushkal and Lee, and depends on the validity of the Novikov conjecture for such fundamental groups.

Keywords:

four-manifolds, homotopy type, obstruction theory, homology with local coefficients, surgery, normal invariant, assembly map four-manifolds, homotopy type, obstruction theory, homology with local coefficients, surgery, normal invariant, assembly map

MSC Classifications:

57N65, 57R67, 57Q10 show english descriptions Algebraic topology of manifolds
Surgery obstructions, Wall groups [See also 19J25]
Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. [See also 19B28] 57N65 - Algebraic topology of manifolds
57R67 - Surgery obstructions, Wall groups [See also 19J25]
57Q10 - Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. [See also 19B28]

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