The $E_8$-boundings of homology spheres and negative sphere classes in $E(1)$

Motoo Tange

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arXiv · 1408.5528

The $E_8$-boundings of homology spheres and negative sphere classes in $E(1)$

Motoo Tange

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Abstract

We define invariants $\frak{ds}$ and $\overline{\frak{ds}}$, which are the maximal and minimal second Betti number divided by $8$ among definite spin boundings of a homology sphere. The similar invariants $g_8$ and $\overline{g_8}$ are defined by the maximal (or minimal) product sum of $E_8$-form of bounding 4-manifolds. We compute these invariants for some homology spheres. We construct $E_8$-boundings for some of Brieskorn 3-spheres $\Sigma(2,3,12n+5)$ by handle decomposition. As a by-product of the construction, some negative classes which consist of addition of several fiber classes plus one sectional class in $E(1)$ are represented by spheres.

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BibTeX RIS

Motoo Tange. 2014-08-23. The $E_8$-boundings of homology spheres and negative sphere classes in $E(1)$. https://arxiv.org/abs/1408.5528

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