Cancellation for 4-manifolds with virtually abelian fundamental group
| Autor: | Khan, Qayum |
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| Rok vydání: | 2016 |
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| Zdroj: | Topology and its Applications, Volume 220 (2017), 14--30 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.topol.2017.01.013 |
| Popis: | Suppose $X$ and $Y$ are compact connected topological 4-manifolds with fundamental group $\pi$. For any $r \geqslant 0$, $Y$ is $r$-stably homeomorphic to $X$ if $Y \# r(S^2 \times S^2)$ is homeomorphic to $X \# r(S^2\times S^2)$. How close is stable homeomorphism to homeomorphism? When the common fundamental group $\pi$ is virtually abelian, we show that large $r$ can be diminished to $n+2$, where $\pi$ has a finite-index subgroup that is free-abelian of rank $n$. In particular, if $\pi$ is finite then $n=0$, hence $X$ and $Y$ are $2$-stably homeomorphic, which is one $S^2 \times S^2$ summand in excess of the cancellation theorem of Hambleton--Kreck. The last section is a case-study investigation of the homeomorphism classification of closed manifolds in the tangential homotopy type of $X = X_- \# X_+$, where $X_\pm$ are closed nonorientable topological 4-manifolds whose fundamental groups have order two. Comment: 14 pages |
| Databáze: | arXiv |
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