A Result in Surgery Theory

Abstract We study the topological 4-dimensional surgery problem for a closed connected orientable topological 4-manifold $X$ with vanishing second homotopy and ${{\pi }_{1}}\left( X \right)\,\cong \,A\,*\,F\left( r \right)$ , where $A$ has one end and $F\left( r \right)$ 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.

[1]  A. Cavicchioli,et al.  Topology of four-manifolds with special homotopy groups , 2006, Bulletin of the Australian Mathematical Society.

[2]  F. Spaggiari.  On the stable classification of spin four-manifolds , 2003 .

[3]  F. Spaggiari.  Four-manifolds with π1–free second homotopy , 2003 .

[4]  F. Quinn,et al.  CONTROLLED SURGERY WITH TRIVIAL LOCAL FUNDAMENTAL GROUPS , 2001, math/0111269.

[5]  J. Ratcliffe,et al.  On the Davis hyperbolic 4-manifold , 2001 .

[6]  Ronnie Lee,et al.  Surgery on closed 4-manifolds with free fundamental group , 2001, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  F. Quinn,et al.  Subexponential groups in 4-manifold topology , 2000, math/0001063.

[8]  Hans-Joachim Baues,et al.  Homotopy Type and Homology , 1996 .

[9]  Dušan D. Repovš,et al.  On the stable classification of certain 4-manifolds , 1995, Bulletin of the Australian Mathematical Society.

[10]  M. Freedman,et al.  4-Manifold topology I: Subexponential groups , 1995 .

[11]  M. Freedman,et al.  4-Manifold topology II: Dwyer's filtration and surgery kernels , 1995 .

[12]  J. Hillman.  Free Products and 4‐Dimensional Connected Sums , 1995 .

[13]  J. Hillman.  The Algebraic Characterization of Geometric 4-Manifolds , 1994 .

[14]  Andrew Ranicki,et al.  Algebraic L-theory and Topological Manifolds , 1993 .

[15]  M. Kreck,et al.  On the classification of topological 4-manifolds with finite fundamental group , 1988 .

[16]  Michael W. Davis.  A hyperbolic 4-manifold , 1985 .

[17]  I. Chiswell.  Exact sequences associated with a graph of groups , 1976 .

[18]  Sergei Novikov,et al.  Novikov conjecture , 2010, Scholarpedia.

[19]  S. Ferry,et al.  Epsilon Surgery Theory , 2006 .

[20]  Bj,et al.  Three-dimensional surgery theory , UNil-groups and the Borel conjecture , 2003 .

[21]  S. Roushon.  L-Theory of 3-manifolds with nonvanishing first Betti number , 2000 .

[22]  Dušan D. Repovš,et al.  Four-Manifolds With Surface Fundamental Groups , 1997 .

[23]  T. Matumoto,et al.  On 4-dimensional closed manifolds with free fundamental groups , 1995 .

[24]  S. Ferry,et al.  Novikov Conjectures, Index Theorems and Rigidity: A history and survey of the Novikov conjecture , 1995 .

[25]  A. Cavicchioli,et al.  On 4-manifolds with free fundamental group , 1994 .

[26]  Michael H. Freedman,et al.  Topology of 4-manifolds , 1990 .

[27]  S. Cappell.  Mayer-vietoris sequences in hermitian K-theory , 1973 .

[28]  F. Quinn.  A geometric formulation of surgery , 1970 .

[29]  Charles Terence Clegg Wall,et al.  Surgery on compact manifolds , 1970 .