## Round handles, logarithmic transforms and smooth 4-manifolds.(English)Zbl 1297.57064

An $$m$$-dimensional round $$k$$-handle is $$S^{1}$$ times an $$(m-1)$$-dimensional $$k$$-handle, that is an $$S^{1} \times D^{k} \times D^{m-k-1}$$ attached along $$S^{1} \times \partial D^{k} \times D^{m-k-1}$$. A logarithmic transform of a $$4$$-manifold is analogous to a Dehn surgery on a 3-manifold except that is is performed using an embedded $$2$$-torus with trivial normal bundle in the $$4$$-manifold. R. J. Stern posed [in: D. A. Ellwood (ed.) et al., Floer homology, Gauge theory, and low-dimensional topology. Proceedings of the Clay Mathematics Institute 2004 summer school, Budapest, Hungary, June 5–26, 2004. Providence, RI: American Mathematical Society (AMS). Cambridge, MA: Clay Mathematics Institute. Clay Mathematics Proceedings 5, 225–239 (2006; Zbl 1106.57024)] the following two questions 1. Are any two arbitrary closed smooth oriented simply connected $$4$$-manifolds $$X$$ and $$X'$$ in the same homeomorphism class related by a sequence of logarithmic transforms along tori? and 2. Is there a cobordism between $$X$$ and $$X'$$ which is composed of round $$2$$-handles only? This paper answers these two questions in the affirmative. Further the paper shows that if the four manifold $$X_{m+1}$$ is obtained from $$X_{m}$$ by a logarithmic transform then $$X_{m+1}$$ and $$X_{m}$$ become diffeomorphic after one stabilization by taking a connected sum with $$S^{2} \times S^{2}$$. The proofs rely on geometric arguments.

### MSC:

 57R50 Differential topological aspects of diffeomorphisms 57R65 Surgery and handlebodies 57R90 Other types of cobordism

Zbl 1106.57024
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