3.1 The definition

Definition 3.1   An n-singular integral homology sphere is a pair (M,L) where M is an integral homology sphere and $L=\bigcup_{i=1}^n L_i$ is a unit-framed algebraically split ordered n-component link in M. Namely, the components L_i of M are numbered 1 to n(``ordered''), framed with $\pm 1$ framing (``unit framed''), and the pairwise linking numbers between the different components of L are 0 (``algebraically split''). We think of L as marking n sites for performing small modifications of M, each modification being the surgery on one of the components of L. Let us temporarily define ${\mathcal M}_n$to be the ${\mathbb Z}$-module of all formal ${\mathbb Z}$-linear combinations of n-singular integral homology spheres. A correction to the definition of ${\mathcal M}_n$ will be given in Definition 3.2 below. Notice that ${\mathcal M}_0$, which we often simply denote by ${\mathcal M}$, is simply the space of all ${\mathbb Z}$-linear combinations of integral homology spheres.

If $L=L^1\cup L^2$ is a framed link (presented as a union of two sublinks L¹ and L²) in some 3-manifold M, we denote by (M,L¹)_L²the result of surgery² of (M,L¹) along L². Namely, (M,L¹)_L² is a pair (M',L¹'), in which M' is the result of surgery of M along L², and L¹'is the image in M' of L¹. Notice that if (M,L) is an (n+1)-singular integral homology sphere, then (M,L-L_i)_Li is again an n-singular integral homology sphere for any component L_i of L.

We now wish to define the co-derivative map $\delta:{\mathcal M}_{n+1}\to{\mathcal M}_n$, whose adjoint will be the differentiation map for invariants:

Definition 3.2   Define $\delta_i$ on generators by $\delta_i(M,L)=(M,L-L_i)-(M,L-L_i)_{L_i}$, and extend it to be a ${\mathbb Z}$-linear map ${\mathcal M}_{n+1}\to{\mathcal M}_n$. For later convinience, we want to set $\delta=\delta_i$ for any i, but the different i's may give different answers. We resolve this by redefining ${\mathcal M}_n$. Set

$${\mathcal M}_n=\left(\text{old }{\mathcal M}_n\right) \lef... ...gular integral homology spheres \end{center}}\right)\right..$$ (5)

We can now set (in the new ${\mathcal M}_n$)

$$\delta(M,L)=(M,L-L_i)-(M,L-L_i)_{L_i} \quad \text{for {\em any} }i.$$

The relations in equation (5) are called ``the co-differentiability relations''.

We can finally differentiate invariants using the adjoint $\partial=\delta^\star:{\mathcal M}_n^\star\to{\mathcal M}_{n+1}^\star$. That is, if $I\in{\mathcal M}_n^\star$ is a differentiable invariant of n-singular integral homology spheres (namely, which vanishes on the co-differentiability relations), let its derivative $I'\in{\mathcal M}_{n+1}^\star$be $\partial I=I\circ\delta$. Iteratively, one can define multiple derivatives such us I^(k) for any $k\geq 0$.

Definition 3.3   (Ohtsuki [Oh1] We say that an invariant I of integral homology spheres is of type nif its n+1st derivative vanishes. We say that it is of finite type if it is of type n for some natural number n.

Unravelling the definitions, we find that I is of type n precisely when for all integral homology spheres M and all unit-framed algebraically split (n+1)-component links L in M,

$$\sum_{L'\subset L}(-1)^{\vert L'\vert}I\left(M_{L'}\right) = 0,$$ (6)

where the sum runs on all sublinks L' of L (including the empty and full sublinks), |L'| is the number of components of L', and M_L' is the result of surgery of M along L'. We will not use equation (6) in this paper.

Footnotes

... surgery²

We recall some basic facts about surgery in Section 3.2.1.