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The geography problem for 4-manifolds with specified fundamental group


Authors: Paul Kirk and Charles Livingston
Journal: Trans. Amer. Math. Soc. 361 (2009), 4091-4124
MSC (2000): Primary 57M05, 57N13, 57R19
DOI: https://doi.org/10.1090/S0002-9947-09-04649-2
Published electronically: March 16, 2009
MathSciNet review: 2500880
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Abstract: For any class $ \mathcal{M}$ of 4-manifolds, for instance the class $ \mathcal{M}(G)$ of closed oriented manifolds with $ \pi_1(M) \cong G$ for a fixed group $ G$, the geography of $ \mathcal{M}$ is the set of integer pairs $ \{(\sigma(M), \chi(M)) \vert M \in \mathcal{M}\}$, where $ \sigma$ and $ \chi$ denote the signature and Euler characteristic. This paper explores general properties of the geography of $ \mathcal{M}(G)$ and undertakes an extended study of $ \mathcal{M}(\mathbf{Z}^n)$.


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Additional Information

Paul Kirk
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: pkirk@indiana.edu

Charles Livingston
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: livingst@indiana.edu

DOI: https://doi.org/10.1090/S0002-9947-09-04649-2
Keywords: Hausmann-Weinberger invariant, fundamental group, four-manifold, minimal Euler characteristic, geography
Received by editor(s): May 25, 2007
Published electronically: March 16, 2009
Additional Notes: This work was supported by grants from the NSF
Article copyright: © Copyright 2009 American Mathematical Society

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