Space of smooth 1-knots in a 4-manifold:
is its algebraic topology sensitive to smooth structures?

One of the most powerful invariants, which distinguish smooth structures on a 4-manifold, is the Seiberg–Witten invariant. It can be presented in several forms. In particular, the Seiberg–Witten invariant of a smooth closed simply-connected 4-manifold $X$ can be identified with an element ${SW}_{X}$ of $\mathbb{Z}[H_{2}(X)]$, the integer group algebra of the second homology group $H_{2}(X)$.

This identification makes the Seiberg–Witten invariant resembling the Alexander polynomial of a link. For a classical link $L\subset S^{3}$, the Alexander polynomial is an element of $\mathbb{Z}[H_{1}(S^{3}\smallsetminus L)]$, the integer group algebra of the first homology group.

The Alexander polynomial can be defined in many ways. In particular, it admits purely topological definitions.

The similarity between Seiberg–Witten invariant and Alexander polynomial gives a hope to find either a definition of Seiberg–Witten invariant free of heavy analysis, or, at least, to invent similar invariants that solve the same problems, but are defined and calculated in ways more traditional for topology. This hope is supported by a relation between the Seiberg–Witten invariant and the Alexander polynomial discovered by Fintushel and Stern [1998] .

Let $X$ be any simply connected closed smooth 4-manifold. Suppose that $X$ contains a smoothly embedded torus $T$ with simply connected complement $X\smallsetminus T$ and with zero self-intersection number $T\circ T$. Since $T\circ T=0$, the normal bundle of $T$ is trivial, and a tubular neighborhood of $T$ can be identified with $T\times D^{2}$.

A knot surgery on $T$ takes away from $X$ the interior of a tubular neighborhood $T\times D^{2}$ of $T$ and attaches $S^{1}\times(S^{3}\smallsetminus N_{K})$ instead, where $N_{K}$ is an open tubular neighborhood of a smooth knot $K\subset S^{3}$. Observe that the boundary of $S^{3}\smallsetminus N_{K}$ is diffeomorphic to the 2-torus $S^{1}\times S^{1}$, and hence the boundary of the inserted piece $S^{1}\times(S^{3}\smallsetminus N_{K})$ is diffeomorphic to the 3-torus $S^{1}\times S^{1}\times S^{1}$ as well as the boundary $T\times\partial D^{2}$ of the piece removed. The attaching is performed by a diffeomorphism $S^{1}\times\partial(S^{3}\smallsetminus N_{K})\to T\times\partial D^{2}$ which maps $\{pt\}\times l$, where $l$ is a longitude, i.e., a circle bounding in $S^{3}\smallsetminus K$, to a fiber $\{pt\}\times\partial D^{2}$. This requirement on the attaching map does not determine the map up to diffeotopy, and hence does not necessarily determine $X_{K}$ up to diffeomorphism. However, by the following theorem, under some assumptions, all the manifolds obtained from the same $(X,T)$ and $K\subset S^{3}$ by knot surgeries have the same Seiberg–Witten invariant.

Present definitions of the Seiberg–Witten invariant are not applicable to $(S^{3}\smallsetminus K)\times S^{1}$ or $(S^{3}\smallsetminus N_{K})\times S^{1}$. So, one cannot speak about ${SW}_{(S^{3}\smallsetminus K)\times S^{1}}$. However, if the ${SW}$ was extended to this setup and satisfied a reasonable additivity property, Fintushel–Stern Theorem would suggests that ${SW}_{(S^{3}\smallsetminus K)\times S^{1}}$ should be $\Delta_{K}$.

The Alexander polynomial of a classical knot $K$ is the order of $\mathbb{Z}[H_{1}(S^{3}\smallsetminus K)]$-module $H_{1}(\widetilde{S^{3}\smallsetminus K})$, where $\widetilde{S^{3}\smallsetminus K}\to S^{3}\smallsetminus K$ is the infinite cyclic covering. The automorphism group of this covering is $H_{1}(S^{3}\smallsetminus K)=\mathbb{Z}$, it acts in the homology group $H_{1}(\widetilde{S^{3}\smallsetminus K})$, turning it into a $\mathbb{Z}[\mathbb{Z}]=\mathbb{Z}[t,t^{-1}]$-module.

This module is called the Alexander module of $K$. It is finitely generated and admits a square matrix of relations. The determinant of this matrix is the Alexander polynomial. It is defined up to multiplication by units of $\mathbb{Z}[\mathbb{Z}]$ (that is by monomials $\pm t^{k}$).

According to this construction, the Alexander polynomial happens to belong to $\mathbb{Z}[H_{1}(S^{3}\smallsetminus K)]$. The Seiberg–Witten invariant of $X$ belongs to $\mathbb{Z}[H_{2}(X)]$. What could be a space $Y$ such that $H_{1}(Y)=H_{2}(X)$?

There is an obvious candidate for such $Y$, the loop space $\Omega X$ of $X$. Indeed, $\pi_{i}(\Omega X)=\pi_{i+1}(X)$, and, in particular, $\pi_{1}(\Omega X)=\pi_{2}(X)$; in the case of simply connected $X$, $\pi_{2}(X)$ is isomorphic to $H_{2}(X)$ by the Hurewicz theorem. Thus $\pi_{1}(\Omega X)=H_{2}(X)$. Therefore $\pi_{1}(\Omega X)$ is commutative, and hence $H_{1}(\Omega X)=\pi_{1}(\Omega X)=H_{2}(X)$.

However, the loop space per se cannot do the job:

• •

  First, the homotopy type of $\Omega X$ depends only on the homotopy type of $X$. Therefore homotopy invariants of $\Omega X$ cannot distinguish smooth structures on $X$.

• •

  Second, $\Omega X$ is an $H$-space. Therefore $\pi_{1}(\Omega X)$ acts on $H_{*}(\widetilde{\Omega X})$ trivially.

How to improve $\Omega X$?

First, let us make it closer to the smooth structure of $X$. The loop space $\Omega X$ contains the space $\Omega_{Diff}X$ of differentiable loops $S^{1}\to X$ that have at the base point fixed non-vanishing differential. The replacement of $\Omega X$ by $\Omega_{Diff}X$ does not change the homotopy type: well-known approximation theorems imply that $\Omega_{Diff}X$ is a deformation retract of $\Omega X$.

Let $KX\subset\Omega_{Diff}X$ be the subspace which consists of loops that are smooth embeddings. Denote $\Omega_{Diff}X\smallsetminus KX$ by $DX$. Observe that $\operatorname{codim}_{\Omega_{Diff}X}DX=2$, hence the inclusion homomorphism $\pi_{1}(KX)\to\pi_{1}(\Omega_{Diff}X)=\pi_{1}(\Omega X)$ is onto.

Obvious suggestion: consider the covering $\widetilde{KX}\to KX$ induced by the universal covering $\widetilde{\Omega X}\to\Omega X$. The automorphism group of the covering $\widetilde{\Omega X}\to\Omega X$ is $H_{1}(\Omega X)=H_{2}(X)$. Therefore $H_{*}(\widetilde{\Omega X})$ and $H_{*}(\widetilde{KX})$ are modules over $\mathbb{Z}[H_{1}(\Omega X)]=\mathbb{Z}[H_{2}(X)]$.

The action of $H_{1}(\Omega X)$ in $H_{*}(\widetilde{\Omega X})$ is trivial, since $\Omega X$ is an $H$-space, while $KX$ is not an $H$-space and $H_{*}(\widetilde{KX})$ may be an interesting $\mathbb{Z}[H_{2}(X)]$-module. It has a broad range of invariants belonging to $\mathbb{Z}[H_{2}(X)]$ similar to the Alexander polynomial. Indeed, if $X$ is simply connected, then $H_{2}(X)$ is a free abelian group of finite rank $r$, and $\mathbb{Z}[H_{2}(X)]$ is isomorphic to the ring $\Lambda_{r}=\mathbb{Z}[t_{1},t_{1}^{-1},t_{2},t_{2}^{-1},\dots t_{r},t_{r}^{-1}]$ of Laurent polynomial in $r$ variables with integer coefficients, the same ring as in the situation of the Alexander module of a classical link.

A finitely generated module $M$ over $\Lambda_{r}$ gives rise to a filtration of $\mathbb{Z}[H_{2}(X)]$

by Fitting ideals. The $i$th Fitting ideal $\operatorname{Fitt}_{i}(M)$ is generated by the minors (determinants of submatrices) of order $r-i$ of the matrix of defining relations for $M$ In the topological literature Fitting ideals are called also elementary ideals. A generator of the minimal principal ideal containing $\operatorname{Fitt}_{i}(M)$ is denoted by $\Delta_{i}(M)$.

In the context of link theory, when $\Lambda_{r}=H_{1}(S^{3}\smallsetminus L)$, where $L$ is an $r$-component link and $M=H_{1}(\widetilde{S^{3}\smallsetminus L})$ where $S^{3}\smallsetminus L$ is the maximal abelian covering space of $S^{3}\smallsetminus L$, the Laurent polynomial $\Delta_{i}(M)$ is called the $i$th Alexander polynomial of $L$. The 0th Alexander polynomial is one of the oldest link invariants. It was introduced by Alexander [1928].

Similarly, for a smooth simply connected closed 4-manifold $X$, each $H_{i}(\widetilde{KX})$, as a module over $\mathbb{Z}[H_{2}(X)]$, gives rise to a sequence of Laurent polynomials. They resemble the Seiberg–Witten invariant. At least, they belong to the same $\mathbb{Z}[H_{2}(X)]$. The modules $H_{i}(\widetilde{KX})$ are also invariants of $X$.

The problem: Does there exist homeomorphic smooth simply connected closed 4-manifolds $X_{1}$, $X_{2}$ such that $H_{*}(\widetilde{KX_{1}})$ and $H_{*}(\widetilde{KX_{2}})$ are not isomorphic?

The author discussed the problem with many leading specialists in the field and asked them this question. The answers spread over a broad spectrum. Only one expert expressed a definitely negative opinion. On the other end, there were also very enthusiastic reactions.

The most convincing argument is that for different smooth structures on the same topological manifold $X$ a topologically observable difference between smooth structures was the minimal genus of a smoothly embedded surface realizing an element of $H_{2}(X)$. For any smooth structure each class can be realized by an immersed sphere with transverse self-intersection (by the Hurewicz theorem and transversality), but the minimal number of double point of such immersion depends on the structure. It is greater than or equal to the minimal genus of a smoothly embedded surface realizing the class.

An immersion $f:S^{2}\to X$ can be used to produce a loop in the space of loops $\Omega_{Diff}X$. A loop in $\Omega_{Diff}S^{2}$ sweeping the whole $S^{2}$ composed with $f$ gives a loop in $\Omega_{Diff}X$. Generically this gives a loop in $KX$, because under a generic choice of loop in $\Omega_{Diff}S^{2}$, points in the preimage of a double point of $f$ are passed at different moments. However, in families of loops realizing elements of high-dimensional homology groups $H_{k}(\Omega_{Diff}S^{2})=\mathbb{Z}$ these pairs of points appear on some of the loops. Such a family does not realize a homology class in $KX$ or $\widetilde{KX}$.

How to calculate $H_{i}(\widetilde{KX})$? One may apply Vassiliev’s (1994) idea, which led to discovery of the Vassiliev knot invariants: start the calculation with a study of the dual cohomology, the cohomology of the space of singular knots. The space of singular knots has a rich geometric structure.

The space $DX$ consists of differentiable loops that are not embeddings. It fits to the collection of discriminant hypersurfaces studied by Vassiliev [1994]. The universal covering $\widetilde{\Omega X}\to\Omega X$ defines a covering $\widetilde{DX}\to DX$.

Resolve singularities of the discriminant $DX$, as Vassiliev did. This gives rise to a filtration in $H^{*}(\widetilde{DX})$. The first terms of these filtration are easier to calculate than the whole cohomology group.

At first glance, the situation is much more complicated than in the original setup in the theory of Vassiliev knot invariants. Let us examine the extra difficulties.

First, instead of the space of singular knots, we have to deal with its covering space. What does happen, when we pass from knots to points of a covering space?

In general, if $p:X\to B$ is a covering, $b_{0}\in B$, $x_{0}\in X$ are points such that $p(x_{0})=b_{0}$ and $X$ is path connected, then a point $x\in X$ is uniquely determined by its image $p(x)\in B$, a path $s:I\to B$ such that $s(0)=b_{0}$, and its covering path $\widetilde{s}:I\to X$ starts at $x_{0}$ and finishes at $x$. The path $s$ is defined up to path homotopy and multiplying by loops covered by loops in $X$.

In particular, a point of $\widetilde{\Omega X}$ (no matter if it belongs to $\widetilde{KX}$ or $\widetilde{DX}$) is defined by a loop $u:S^{1}\to X$ and a continuous map $f:D^{2}\to X$ with $f|_{S^{1}}=u$ considered up to homotopy which is fixed on $S^{1}$. The action of $H_{2}(X)$ in $\widetilde{\Omega X}$ is realized in this model as addition to the homotopy class of $f$ homotopy classes of maps $S^{2}\to X$ realizing elements of $H_{2}(X)$.

Second, the space of all loops in the Vassiliev setup is contractible (Recall that loops there have the base point and the tangent vector at the base point fixed). This allowed to apply the Alexander duality between the homology of the discriminant and cohomology of its complement, the space of knots. This was done in a finite dimensional (say, polynomial) approximation of the loop space.

Here the loop space is not contractible. Therefore, the Alexander duality is not applicable. However, it may be replaced by the Alexander–Poincaré duality between the homology of $\widetilde{KX}$ and a relative cohomology, the cohomology of $(\widetilde{\Omega_{Diff}X},\widetilde{DX})$.

The homology of loop space $\Omega_{Diff}X$ and its universal covering accommodate a rich structure of the string topology operations (Chas and Sullivan [1999]). Geometrically, one can expect that this structure incorporates the same information as the natural filtration of the discriminant and the Alexander–Poincaré duality. Apparently the connection have not been investigated. Nonetheless, it would be natural to expect that the string topology is as strong as the invariants discussed above.

Problem. Is the string topology sensitive to smooth structures on 4-manifolds?

It would be interesting also to find direct relations between the Vassiliev invariants theory and the string topology in the setup of classical knot theory.