In the late 1980's, it was shown that the Casson invariant appears in the difference between the two filtrations of the Torelli group: the lower central series and the Johnson filtration. This was interpreted as the secondary characteristic class $d_1$ associated with the fact that the first MMM class vanishes on the Torelli group. It is a rational generator of $H^1(\mathcal{K}_g;\mathbb{Z})^{\mathcal{M}_g}\cong\mathbb{Z}$ where $\mathcal{K}_g$ denotes the Johnson subgroup of the mapping class group $\mathcal{M}_g$. Then Hain proved, as a particular case of his fundamental result, that this is the only difference in degree 2. In this paper, we prove that no other invariant than the above gives rise to new rational difference between the two filtrations up to degree 6. We apply this to determine $H_1(\mathcal{K}_g;\mathbb{Q})$ explicitly by computing the description given by Dimca, Hain and Papadima. We also show that any finite type rational invariant of homology 3-spheres of degrees up to 6, including the second and the third Ohtsuki invariants, can be expressed by $d_1$ and lifts of Johnson homomorphisms.

中文翻译：

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Torelli 群、Johnson 核和同源球的不变量

在 1980 年代后期，研究表明 Casson 不变量出现在 Torelli 组的两个过滤之间的差异中：下中央级数和 Johnson 过滤。这被解释为与第一个 MMM 类在 Torelli 组上消失的事实相关的次要特征类 $d_1$。它是 $H^1(\mathcal{K}_g;\mathbb{Z})^{\mathcal{M}_g}\cong\mathbb{Z}$ 的有理生成器，其中 $\mathcal{K}_g$表示映射类群 $\mathcal{M}_g$ 的 Johnson 子群。然后，Hain 证明，作为他的基本结果的一个特例，这是 2 阶的唯一差异。 在本文中，我们证明除了上述不变量之外，没有其他不变量会导致两个过滤之间的新的有理差异达到 6 阶. 我们应用它来确定 $H_1(\mathcal{K}_g; \mathbb{Q})$ 通过计算 Dimca、Hain 和 Papadima 给出的描述来明确。我们还表明，任何有限类型的同调 3 球度数高达 6 的有理不变量，包括第二个和第三个 Ohtsuki 不变量，都可以用 $d_1$ 和 Johnson 同态的提升来表示。