The E₈-boundings of homology spheres and negative sphere classes in E(1)

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• 作者：Motoo Tange¹ ; ^{tange@math.tsukuba.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：57R55 ; 57R65
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：202
• 期：Complete
• 页码：160-182
• 全文大小：585 K

文摘

We define topological invariants of homology 3-sphere, src">ds$\mathfrak{ds}$ and src">src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166864115006008-si132.gif">$\underset{\_}{\mathfrak{ds}}$, which are the maximal and minimal second Betti number divided by 8 among definite spin boundings of the homology sphere. We also define similar invariants src">g₈$g_8$ and src">src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166864115006008-si135.gif">$\underset{\_}{g_8}$ by the maximal (or minimal) product sum of the quadratic form src">E₈$E_8$ of bounding 4-manifolds. The aim of these invariants is to measure the size of bounding definite spin 4-manifold. We give several ways to construct definite spin boundings. In particular, we construct uncommon src">E₈$E_8$-boundings for src">Σ(2,3,12n+5)$\mathrm{\Sigma }(2,3,12n+5)$ by using handle decomposition. As a by-product of this construction, we show that some negative 2nd homology classes src">k[f]−[s]$k[f]-[s]$ in src">E(1)$E(1)$ are represented by a sphere, where f and s   are a fiber and sectional class of src">E(1)$E(1)$.