Homology, Homotopy and Applications, vol.5(1), 2003, pp.251–260

ON SIGNATURES AND A SUBGROUP OF

A CENTRAL EXTENSION TO THE MAPPING CLASS GROUP

JONATHAN NATOV

(communicated by Jonathan Rosenberg)

Abstract Atiyah’s work [1] describes the relationship between multiplication in a central extension of the mapping class group of a surface of genus n and the signatures of 4-dimensional manifolds. This work studies a subgroup of the central extension, which comes from the image of a representation of the pure framed braid group on n-strands found in [5], and the signatures of corresponding 4-manifolds via a split exact sequence. We construct a splitting map to prove the sequence is split exact, and we use the splitting to give a topological description of homology classes in 4-dimensional manifolds with non-zero intersection. We conclude with a description of multiplication in the subgroup.

1. Introduction

In his paper [1], M. Atiyah uses 2-framings to place Witten’s 3-manifold invariant in the context of algebraic topology. Witten’s invariant relies on 2-framings, which are related to a central extension \overline{Γ_{n}} of the mapping class group, Γ_{n}, of a compact oriented surface of genus n. In [1] Atiyah proves the sequence

0 → ℤ → \overline{Γ_{n}} → Γ → 0

is an exact sequence. \overline{Γ_{n}} consists of pairs (f, α) with f ∈ Γ_{n}, and α a 2-framing on a 3-manifold.

Given an oriented compact 3-manifold Y with tangent bundle T_{Y}, Atiyah defined a 2-framing as a homotopy class of trivializations of T_{Y} ⊕ T_{Y} viewed as a Spin(6) bundle of the diagonal embedding SO(3) → SO(3) × SO(3) → SO(6). He then showed the correspondence between a 2-framing α and an integer via

σ(α) = Sign(Z) − \frac{1}{6}p_{1}(T_{Y} ⊕ T_{Y}),

where Z is a 4-manifold with boundary Y, and p_{1} is the relative Pontrjagin class p_{1} ∈ H^{4}(Y × I, Y × ∂I). Atiyah proved σ does not depend on the choice of Z.

supported by PSC-CUNY grant 669430. Received October 16, 2001, revised July 16, 2003; published on July 23, 2003. 2000 Mathematics Subject Classification: 57N05, 57M25, 20F36, 57N13. Key words and phrases: Mapping Class Group, Pure Braids, Signatures. \begin{matrix}c \\ ◯ \end{matrix} 2003, Jonathan Natov. Permission to copy for private use granted.

In this paper we use a representation Φ, see section 2, from the pure framed braid group ℤ^{n} ⊕ P_{n} to the mapping class group Γ_{n}, and study the pullback of the image subgroup in \overline{Γ_{n}}, which we denote \overline{ℤ^{n} ⊕ P_{n}}.

0 ⟶ ℤ ⟶ \overline{Γ_{n}} ⟶ Γ_{n} ⟶ 0
↑  ↑  ↑  ↑^{Φ}  ↑
0 ⟶ ℤ ⟶ \overline{ℤ^{n} ⊕ P_{n}} ⟶ ℤ^{n} ⊕ P_{n} ⟶ 0

The 4-manifolds in [1] are constructed using mapping tori X_{f} and X_{g}, where X is a surface (see definition 3). To explain restricting to \overline{ℤ^{n} ⊕ P_{n}}, in [5] it was shown that any compact orientable 3-manifold can be represented as X_{f}, where f ∈ im(Φ) and X is compact oriented surface of genus n. Furthermore, the multiplication in \overline{ℤ^{n} ⊕ P_{n}} is computed via linking matrices coming from the closures of pure framed braids, see theorem 2 of section 5.

We use the above correspondence between 2-framings and integers to consider elements of \overline{ℤ^{n} ⊕ P_{n}} to be pairs (h, c), consisting of mapping class elements h, and integers c. The group multiplication is given by (f, a) · (g, b) = (fg, a + b + 3 · Sign(W_{f,g}^{4})), where Sign(W_{f,g}^{4}) is the signature of the intersection form

H_{2}(W_{f,g}^{4}, ℤ) × H_{2}(W_{f,g}^{4}, ℤ) → ℤ.

H_{2}(W_{f,g}^{4}) is computed using a Mayer Vietoris sequence, see equation 1 of section 3. We construct a splitting map τ of section 4 to show the Mayer Vietoris sequence is split exact. In section 4 we use τ to construct specific examples of classes in H_{2}(W_{f,g}^{4}) with non zero intersection. Note: we compute homology groups over the integers.

2. The representation Φ : ℤ^{n} ⊕ P_{n} → Γ_{n}

Let B_{n}, n > 1, be the group given by generators σ_{1}, …, σ_{n−1} and relations

1) σ_{i}σ_{j} = σ_{j}σ_{i} for |i − j| ⩾ 2

2) σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}

B_{n} is called the Braid Group on n-strands. Let Σ_{n} denote the symmetric group acting on {1, 2, ⋯, n}, and π : B_{n} → Σ_{n} denote the homomorphism sending σ_{i} to the transposition (i, i + 1). Let P_{n} denote the kernel of π.

Definition 1. The pure framed braid group on n-stands is the direct sum ℤ^{n} ⊕ P_{n}, where ℤ^{n} denotes the group of n − tuples of integers under addition. a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} a_{1} + b_{1} a_{2} + b_{2} a_{3} + b_{3}

f  g  fg

Figure 1: Multiplication of pure framed braids.

We will use a geometric description of pure framed braids. The equivalence between the algebraic and geometric definitions is found in [2]. Consider a cylinder D^{2} × [0, 1] with projection π_{1} : D^{2} × [0, 1] → D^{2} defined π_{1}(x, t) = x. Choose n non-intersecting (smooth) curves in D^{2} ×[0, 1], where each curve intersects D^{2} ×{t} in exactly one point, for 0 ⩽ t ⩽ 1. If x_{i} and y_{i} denote the points the i^{th} curve intersects D^{2} × {0} and D^{2} × {1} respectively, then π_{1}(x_{i}) = π_{1}(y_{i}) for 1 ⩽ i ⩽ n. We require the x_{i} to be evenly spaced along a fixed diameter in the interior of D^{2} ×{0}. In Figure 1 we see the projection of pure framed braids on 2 strands into the plane. We use breaks in the curves to indicate one strand passing over another. The framed pure braid has an integer assigned to each curve called the framing number. Two pure framed braids are equivalent if they are isotopic via an isotopy of D^{2} × [0, 1] which fixes the end-points of the curves and preserves the framings. We illustrate the multiplication in figure 1 with (a_{1}, a_{2}, a_{3})f and (b_{1}, b_{2}, b_{3})g ∈ Z^{3} ⊕ P_{3}. Their product (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3})fg is formed by stacking f on top of g and adding the corresponding framings. Note: we omit a framing number if it is zero.

Notation 1. By D_{n} we mean a (unit) disc D^{2} with n open disjoint discs removed from the interior. The removed discs were chosen so that their centers were evenly spaced on a fixed diameter of D^{2}.

Let pfb(D_{n}) be the group of isotopy classes of homeomorphisms from D_{n} to D_{n} which fix the ∂D_{n} pointwise.

Theorem 1. (Prasolov, V. V. and Sossinsky, A. B.) pfb(D_{n}) is isomorphic to ℤ^{n} ⊕ P_{n}.

The proof appears in [6]. To give a description of the isomorphism, let [g] ∈ pfb(D_{n}), and g_{0} be a representative. To obtain a pure framed braid corresponding to [g], we extend the map g_{0} to g : D^{2} → D^{2} by g(x) = x for x not in the interior of D_{n}. This extension is possible because g_{0}(x) = x for x ∈ ∂D_{n}. Now define an isotopy I : D^{2} × [0, 1] → D^{2},

I(x, t) = \begin{cases}x, & if t = 0; \\ x, & if |x| > t; \\ tg(t^{−1}x), & otherwise. \end{cases}

Note I(x, 0) = x and I(x, 1) = g(x). In the complement of the interior of D_{n} in D^{2} there are n closed discs. If c_{i} is the center of the i^{th} disc, then (I(c_{i}, t), t) for 0 ⩽ t ⩽ 1 is an arc in D^{2} × [0, 1]. The n arcs are the n strands of the pure braid. Choose any point b_{i} on the S^{1} boundary of the disc with center c_{i}. The linking number of (I(b_{i}, t), t) with c_{i} × [0, 1] is an integer, and this will be the framing corresponding to the i^{th} strand. For example, in figure 2 we have a Dehn twist about the dashed curve, which is an element of pfb(D_{3}). The pure framed braid is formed by following the image of the interior points c_{1}, c_{2}, c_{3} under the isotopy I.

Definition 2. The mapping class group of a (compact oriented) genus n surface, denoted here by Γ_{n}, is the group of isotopy classes of orientation preserving homeomorphisms from the surface to itself.

An element of pfb(D_{n}) is an isotopy class of maps of D_{n} → D_{n}, or equivalently D_{n} × {1} → D_{n} × {1}. Each representative of the isotopy class, fixes ∂(D_{n} × {1}) pointwise, and may be extended by the identity map to ∂(D_{n} × [0, 1]). This extension is a homomorphism pfb(D_{n}) → Γ_{n}. Composing this map with the isomorphism ℤ^{n} ⊕ P_{n} → pfb(D_{n}) yields Φ : ℤ^{n} ⊕ P_{n} → Γ_{n}. Our viewpoint for the construction of Φ was inspired by [3] in which the authors discussed representations of platts.

Figure 2: A Dehn twist and corresponding pure framed braid.

3. The 4-manifold W_{f,g}^{4}

The construction of W_{f,g}^{4} from [1] requires the definition of a mapping torus.

Definition 3. For a homeomorphism f of a surface X to itself, define the mapping torus X_{f} as \frac{X×[0,1]}{(p,0)∼(f(p),1)}

We are interested in the case where X = ∂(D_{n} × [0, 1]), a surface of genus n, and f a representative of [f] in the image of Φ : ℤ^{n} ⊕ P_{n} → Γ_{n}. We can view X_{f} as a fiber bundle over S^{1}, with fiber ∂(D_{n} × [0, 1]), and X_{f} × [0, 1] as a fiber bundle over an annulus. The 4-manifold W_{f,g}^{4} is defined as the quotient space of (X_{f} × [0, 1]) ∪ (X_{g} × [0, 1]) where, for t ∈ [0.5, 0.75], we identify (∂(D_{n} ×[0, 1])×{t})×{1} of X_{f} ×[0, 1] with (∂(D_{n} ×[0, 1])×{t})×{1} of X_{g} ×[0, 1] by the identity map ∂(D_{n}×[0, 1]) → ∂(D_{n}×[0, 1]). Viewing X_{f} ×[0, 1] and X_{g}×[0, 1] as fiber bundles over annuli, we have identified arcs on the outer boundary circles of the annuli base spaces and their corresponding fiber. We obtain homeomorphic 4-manifolds if different representatives of [f] and [g] are used. The result is a fiber bundle with base D_{2}. Restricting W_{f,g}^{4} to the interior boundary circles of D_{2} we have X_{f} and X_{g}. Restricting to the exterior boundary circle of D_{2} we have X_{fg}.

It is shown in [1] that 4-manifolds Z with ∂Z = X_{f} + X_{g} − X_{fg} have the same signature.

Figure 3: A framed braid and its closure to a framed link.

In order to obtain the signature of W_{f,g}^{4}, we compute the homology via a Mayer Vietoris sequence. This sequence is defined with spaces X_{0} and X_{1}, whose union is W_{f,g}^{4}. The space X_{1} is a fiber bundle over D_{2} with fiber D_{n}. It is constructed in the same way as W_{f,g}^{4}, except that we replace the surface fiber ∂(D_{n} × [0, 1]) with D_{n} × {1}. The fiber in X_{1} over the two interior boundary circles of the base space D_{2} is homeomorphic to complements of tubular neighborhoods of f̂ and of ĝ in solid tori, where f̂ and ĝ denote the closures of the pure braids corresponding to f and g. The closure of a framed braid is formed by connecting the two ends of each strand using a simple closed arc. The result is a link with each path component given the framing coming from the corresponding strand, see figure 3. The space X_{0} is the product space D_{2} × (D_{n} × {0}). Gluing two copies of D_{n} along their boundary is homeomorphic to a genus n surface. If at each point in D_{2} we glue the D_{n}×{1} fiber of X_{1} to the D_{n} × {0} fiber of X_{0}, the result is W_{f,g}^{4}. Using collar neighborhoods of the D_{n} fiber so that X_{0} and X_{1} are open, we obtain the Mayer Vietoris sequence.

\begin{aligned}⋯ & ⟶ & H_{2}(X_{0}) ⊕ H_{2}(X_{1}) & ⟶^{σ} & H_{2}(W_{fg}^{4}) \\ & ⟶^{δ} & H_{1}(X_{0} ∩ X_{1}) & ⟶^{j} & H_{1}(X_{0}) ⊕ H_{1}(X_{1}) & ⟶ & ⋯ \end{aligned} (1)

The map j : H_{1}(X_{0} ∩ X_{1}) → H_{1}(X_{0}) ⊕ H_{1}(X_{1}) takes a → (a, a) and σ(b, c) = b − c. By the exactness of the sequence, im(σ) = ker(δ) and ker(j) = im(δ). In Lemma 2 of section 5 we show that the above sequence is split exact. It will then follow that a basis for H_{2}(W_{fg}^{4}) is isomorphic to a direct sum of a basis for im(σ) with a basis for ker(j).

4. Constructing intersecting classes.

For [f] and [g] in the image of Φ : ℤ^{n} ⊕ P_{n} → Γ_{n} we will define a splitting τ : ker(j) → H_{2}(W_{f,g}^{4}). In lemma 3 we show that only elements in the image of τ can have non-zero intersection. To describe ker(j) we label curves in X_{0} ∩ X_{1}, which is a fiber bundle with base D_{2} and fiber ∂D_{n}; recall D_{n} is a disc D^{2} with interior open discs int(e_{1}), ⋯, int(e_{n}) removed. Label the boundary circles of D_{n} by s_{0}, s_{1}, ⋯, s_{n}, with s_{0} = ∂D^{2}. Let D_{2} be the disc D^{2} with the interior of two discs, int(d_{1}) and int(d_{2}), removed.

Notation 2. Let y_{i} be a point of s_{i} , and x ∈ D_{2}. Set p_{i} = ∂d_{1} × {y_{i}} , q_{i} = ∂d_{2} × {y_{i}} and m_{i} = s_{i} × {x}.

Using the Mayer Vietoris sequence in equation (1), along with the Künneth formula for computing the homology of a product space, one obtains homology bases for H_{1}(X_{0} ∩X_{1}) and H_{1}(X_{0})⊕H_{1}(X_{1}). Write j : H_{1}(X_{0} ∩X_{1}) → H_{1}(X_{0})⊕H_{1}(X_{1}) as j = j_{0} ⊕ j_{1}, so j_{1} : H_{1}(X_{0} ∩ X_{1}) → H_{1}(X_{1}) and j_{0} : H_{1}(X_{0} ∩ X_{1}) → H_{1}(X_{0}). An ordered basis for H_{1}(X_{0} ∩ X_{1}) is {[p_{0}], ⋯, [p_{n}], [q_{0}], ⋯ [q_{n}], [m_{0}], ⋯, [m_{n}]}. An ordered basis for H_{1}(X_{0})⊕H_{1}(X_{1}) is {j_{0}([p_{0}]), j_{0}([s_{0}]), ⋯ j_{0}([s_{n}]), j_{1}([q_{0}]), j_{1}([s_{0}]), ⋯ j_{1}([s_{n}])}. The following lemma is proved with a standard analysis of the matrix of j.

Lemma 1. Let [f], [g] be in the image of Φ : ℤ^{n} ⊕ P_{n} → Γ_{n}. Let F = [F_{ij}] and −G = −[G_{ij}] be the n × n linking matrices of f̂ and −1 times the linking matrix of ĝ, respectively. Set B equal to the submodule of H_{1}(X_{0} ∩ X_{1}) with ordered basis {[p_{1}] − [p_{0}], ⋯, [p_{n}] − [p_{0}], [q_{1}] − [q_{0}], ⋯, [q_{n}] − [q_{0}]}. Define (F − G) : B → H_{1}(X_{0}) ⊕ H_{1}(X_{1}) to be multiplication by the matrix formed by juxtaposing the linking matrices of F and −G.

Then, ker(j) is isomorphic to ker (F −G) ⊕ < [η] >,

where [η] = [m_{0}] − (∑_{i=1}^{n}[m_{i}]).

Example:

Consider the example of representatives (0, 0, 0)f and (−1, 0, 0)g in the image of Φ : ℤ^{3} ⊕ P_{3} → Γ_{3} corresponding to the pure framed braids in figure 1, where the framings a_{1} = a_{2} = a_{3} = b_{2} = b_{3} = 0 and b_{1} = −1. From (0, 0, 0)f and (−1, 0, 0)g we obtain the matrix

(F −G) = (\begin{matrix}0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 & −1 & 0 \end{matrix}) .

Using the symbol T to stand for transpose, two elements in the kernel are

α = (1 0 0 0 0 1)^{T} and β = (0 1 0 −1 0 0)^{T} .

To construct the classes τ(α) and τ(β) we decompose W_{(0,0,0)f,(−1,0,0)g}^{4} into X_{1} and X_{0}. In this case both are fiber bundles with base space D_{2}, and fiber D_{3}. For i = 1, 2, 3, j_{1}([p_{i}]) is a longitude on a tubular neighborhood of the i^{th} strand of the closed braid f̂ in the fiber over ∂d_{1}. Similarly, j_{1}([q_{i}]) corresponds to the i^{th} strand of ĝ in the fiber over ∂d_{2}. The classes j_{0}([p_{i}]) and j_{0}([q_{i}]) are longitudes on tubular neighborhoods of the i^{th} strands of the closed identity braids in the fiber over ∂d_{1} and ∂d_{2}, respectively, in X_{0}. With y_{0} a point on the exterior boundary circle of D_{3}, the classes j_{1}([p_{0}]) and j_{1}([q_{0}]) represent S^{1} × {y_{0}} in the S^{1} × D_{3} fiber over ∂d_{1} and ∂d_{2} in X_{1}, respectively.

Figure 4: τ(α) in X_{1} Figure 5: τ(β) in X_{1}

Since α = ([p_{1}]−[p_{0}])+([q_{3}]−[q_{0}]), attach two annular regions. One annular region has boundary j_{1}([p_{1}]) − j_{1}([p_{0}]), and the second has boundary j_{1}([q_{3}]) − j_{1}([q_{0}]). The first annular region is in the fiber over ∂d_{1} in X_{1}, see the left side of figure 4, corresponding to f̂. To allow the tubular neighborhood with longitude j_{1}([p_{2}]) to pass through, we remove a disc and attach an S^{1} × [0, 1] along the S^{1} × {0} boundary. The annular region with boundary j_{1}([q_{3}]) − j_{1}([q_{0}]) is in the fiber over ∂d_{2}, corresponding to ĝ. We remove a disc and attach a tube so that a tubular neighborhood with longitude j_{1}([q_{2}]) may pass through, see the right side of figure 4. The tubes about j_{1}([p_{2}]) and j_{1}([q_{2}]) are then joined by passing through fibers over D_{2}.

In the fiber over ∂d_{1}, the annular region in X_{1} is glued to the boundary of an annulus in X_{0} between j_{0}([p_{1}]) and j_{0}([p_{0}]). In the fiber over ∂d_{2}, the annular region in X_{1} is glued to an annulus in X_{0} between j_{0}([q_{3}]) and j_{0}([q_{0}]). The surface τ(α) is two tori joined by a tube, and with an orientation is a class of H_{2}(W_{f,g}^{4}). The projection of this class to the base space D_{2} is two circles, concentric to the interior boundary circles joined together with a simple non-intersecting arc, see figure 6. The arc being the projection of the tube joining the annular regions in X_{1}.

Figure 6: The projection of τ(α) and τ(β) to D_{2} in general position.

We construct a second surface τ(β) in a similar fashion. In the fiber over ∂d_{1} in X_{1}, there is an annular region between j_{1}([p_{2}]) and j_{1}([p_{0}]), and a tube is created about a tubular neighborhood with j_{1}([p_{1}]). In the fiber over ∂d_{2} there is an annular region with boundary j_{1}([q_{0}]) and a circle linking j_{1}([q_{1}]) with linking number −1.

This is due to the framing number −1. For the tubular neighborhood with j_{1}([q_{1}]) to pass through the annular region, a disc is removed and a tube is attached, see figure 5. These tubes are joined together. After attaching the boundary to the corresponding annuli in X_{0} we obtain a surface, denoted τ(β). The projection of τ(β) to D_{2} is two circles joined by an arc. τ(α) and τ(β) have an intersection of 1. Without the tubes, the projections to the base space D_{2}, in general position, would be two pairs of concentric circles. Such surfaces would not intersect.

5. The intersection form on H_{2}(W_{f,g}^{4})

Lemma 2. There is a splitting τ : ker(j) → H_{2}(W_{fg}^{4}).

Proof

As in section 4, τ([p_{i}] − [p_{0}]) is an annular region in X_{1} over ∂d_{1} with boundary j_{1}([p_{i}])−j_{1}([p_{0}]) glued to an annulus in X_{0} with boundary j_{0}([p_{i}])−j_{0}([p_{0}]). Define τ([q_{i}]−[q_{0}]) to be an annular region in X_{1} over ∂d_{2} with boundary j_{1}([q_{i}])−j_{1}([q_{0}]) glued to an annulus in X_{0} with boundary j_{0}([q_{i}]) − j_{0}([q_{0}]). τ([η]) is defined to be the surface fiber over a point in D_{2}. To define τ on a general element of ker(j), extend linearly. In the construction of τ(a_{1}([p_{1}]−[p_{0}])+⋯+a_{n}([p_{n}−p_{0}])+b_{1}([q_{1}]−[q_{0}])+⋯+b_{n}([q_{n}]−[q_{0}])), for each a_{k} ≠ 0, there are |a_{k}| copies of annular regions between j_{1}([p_{k}]) and j_{1}([p_{0}]). The algebraic intersection of j_{1}([p_{i}]) with these regions is a_{k}F_{ik}. Similarly, for b_{k} ≠ 0 there are |b_{k}| copies of annular regions between j_{1}([q_{k}]) and j_{1}([q_{0}]), and the algebraic intersection with j_{1}([q_{i}]) is b_{k}G_{ik}. If (a_{1}, ⋯, a_{n}, b_{1}, ⋯, b_{n}) is in ker(F −G), then for 1 ⩽ i ⩽ n, ∑_{k=1}^{n} a_{k}F_{ik} = ∑_{k=1}^{n} b_{k}G_{ik}. Therefore, we can glue | ∑_{k=1}^{n} a_{k}F_{ik}| tubes about j_{1}([p_{i}]) to | ∑_{k=1}^{n} b_{k}G_{ik}| tubes about j_{1}([q_{i}]). By examining the definition of δ one sees that τ is a splitting. Q.E.D.

Lemma 3. Let α, β ∈ H_{2}(W_{f,g}^{4}) and α = a_{1}κ_{1}+a_{2}κ_{2}, where a_{1}, a_{2} ∈ ℤ, κ_{1} ∈ im(τ) and κ_{2} ∈ im(σ) of equation (1). If a_{1} = 0, then the intersection α · β = 0.

Proof

From equation (1) and lemma 2 we have H_{2}(W_{f,g}^{4}) ≈ im(σ)⊕τ(ker(j)). A Mayer Vietoris sequence shows that a class in image of σ can be divided into a class in the fiber over ∂d_{1} of the base D_{2} and a class in the fiber over ∂d_{2}. In general position, the projection of two such classes to D_{2} is a finite set of circles concentric to ∂d_{1} and to ∂d_{2}. Such classes do not intersect. A class in the image of τ projected to the base space D_{2} is a finite set of circles concentric to ∂d_{1} and ∂d_{2}, possibly with some simple arcs between them. In general position, one can arrange the class in the image of σ so its projection to D_{2} is a set of circles interior to the circles coming from the class in the image of τ. Thus, there is no intersection. Q.E.D.

Remark 1. In general position, the fiber over a point does not intersect a surface. Thus, τ([η]) does not contribute to signature computations.

Next we relate the algebraic computation of intersections in W_{f,g}^{4} to our topological description of section 4. In the computation of τ(α) · τ(β),

(\begin{matrix}F & 0_{3×3} \\ 0_{3×3} & −G \end{matrix}) · α = (0 1 0 0 −1 0)^{T}

indicates the tubes created about j_{1}([p_{2}]) and j_{1}([q_{2}]) in the fiber over ∂d_{1} and ∂d_{2}, respectively. The multiplication

β^{T} · ((\begin{matrix}F & 0_{3×3} \\ 0_{3×3} & −G \end{matrix}) · α) =

(0 1 0 −1 0 0) · ((0 1 0 0 −1 0)^{T} ) = 1

reflects that the tube about j_{1}([p_{2}]) intersects the annular region in the fiber over ∂d_{1} with boundary components j_{1}([p_{0}]) and j_{1}([p_{2}]). More generally, we have the next theorem. Recall the map δ : H_{2}(W^{4}) → H_{1}(X_{1} ∩ X_{2}) from equation (1).

Theorem 2. Let [f], [g] be in the image of Φ : ℤ^{n} ⊕ P_{n} → Γ_{n}, and α, β ∈ H_{2}(W_{f,g}^{4}). Set F = [F_{ij}] to be the linking matrix of f̂, and −G = −[G_{ij}] to be −1 times the linking matrix for ĝ. Suppose the projection of δ(α)to ker(F − G) has coordinates (a_{1}, …, a_{n}, b_{1}, …, b_{n}), and the projection of δ(β) has coordinates (c_{1}, …, c_{n}, d_{1}, …, d_{n}). Then, the intersection (α) · (β) =

(c_{1} … c_{n} d_{1} … d_{n}) (\begin{matrix}F & 0_{n×n} \\ 0_{n×n} & −G \end{matrix}) (a_{1} … a_{n} b_{1} … b_{n})^{T}

Proof

By lemma 3 and remark 1, the only non-zero intersections are from classes in the image of τ restricted to ker(F − G). Consider the sum c_{k}(a_{k}F_{1k} + … + a_{k}F_{nk}). When c_{k} ≠ 0 we have |c_{k}| annular regions between with boundary components j_{1}([p_{k}]) and j_{1}([p_{0}]). If a_{k} ≠ 0, there will be |a_{k}F_{ik}| number of tubes around a tubular neighborhood of j_{1}([p_{k}]), formed by avoiding intersections with annular regions between j_{1}([p_{i}]) and j_{1}([p_{0}]). Summing for i = 1 … n yields an algebraic intersection of c_{k}(a_{k}F_{1k} + a_{k}F_{2k} + a_{k}F_{nk}). Accounting for all j_{1}([p_{k}]), 1 ⩽ k ⩽ n, we obtain

(c_{1}, …, c_{n}) (F ) (a_{1} … a_{n})^{T} .

A similar analysis yields the intersection number coming from j_{1}([q_{k}]), 1 ⩽ k ⩽ n, is

− (d_{1}, …, d_{n}) (G) (b_{1} … b_{n})^{T} . Q.E.D.

Using the map Φ : ℤ^{n} ⊕ P_{n} → Γ_{n} we obtain the commutative diagram below.

0 ⟶ ℤ ⟶ \overline{Γ_{n}} ⟶ Γ_{n} ⟶ 0
↑  ↑  ↑  ↑^{Φ}  ↑
0 ⟶ ℤ ⟶ \overline{ℤ^{n} ⊕ P_{n}} ⟶ ℤ^{n} ⊕ P_{n} ⟶ 0

The group \overline{Γ_{n}} is a central extension of the mapping class group Γ_{n} studied in [1], and \overline{ℤ^{n} ⊕ P_{n}} is the pullback of Φ. An element of the subgroup \overline{ℤ^{n} ⊕ P_{n}} is a pair (h, a), where a is an integer and h is in the image of Φ. The multiplication in \overline{ℤ^{n} ⊕ P_{n}} is computed using linking matrices.

In conclusion, we multiply ([(0, 0, 0)f], a) · ([(−1, 0, 0)g], b) in \overline{ℤ^{3} ⊕ P_{3}}. The intersection pairing on W_{fg}^{4} is given by the quadratic form x^{T} Ax where

A = (\begin{matrix}F & 0_{3×3} \\ 0_{3×3} & −G \end{matrix}) = (\begin{matrix}0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 & −1 & 0 \end{matrix})

The signature of the matrix A is 1, and therefore ([(0, 0, 0)f], a) · ([(−1, 0, 0)g], b) = ([(−1, 0, 0)fg], a + b + 3Sign(W_{fg}^{4})) = ([(−1, 0, 0)fg], a + b + 3).

References

[1] Atiyah, M.F. On Framings of 3-Manifolds, Topology, Vol 29, no.1, pp. 1-7, 1990

[2] Birman, J. Braids, Links and Mapping Class Groups, Ann. Math. Stud. 82, 1975

[3] Birman, J., Powell, J. Special Representations for 3-Manifolds, Geometric Topology. Proc. Academic Press, pp. 23-51, 1977

[4] Mayer, W. Die Sigaur von Flachenbundeln, Math. Ann. Vol. 201, pp. 239-264, 1973

[5] Natov, J. Pure Framed Braids and 3-Manifolds, Doctoral Dissertation, Louisiana State Univ. Baton Rouge, LA, 1997

[6] Prasolov, V. V. and Sossinsky, A. B. Knots, Links, Braids and 3-Manifolds: an introduction to new invariants in low dimensional topology, Trans. Math. Monogr., Vol. 154, Amer. Math. Soc., 1996

This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

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Jonathan Natov natovj@mail.com

Department of Mathematics New York City College of Technology, C.U.N.Y. 300 Jay Street Brooklyn, New York 11201