In his 1957 paper, John Milnor introduced link invariants which measure the homotopy class of the longitudes of a link relative to the lower central series of the link group. Consequently, these invariants determine the lower central series quotients of the link group. This work has driven decades of research with profound influence. One of Milnor's original problems remained unsolved: to extract similar invariants from the transfinite lower central series of the link group. We reformulate and extend Milnor's invariants in the broader setting of 3-manifolds, with his original invariants as special cases. We present a solution to Milnor's problem for general 3-manifold groups, developing a theory of transfinite invariants and realizing nontrivial values.
We study the effect of mutation on link concordance and 3-manifolds. We show that the set of links concordant to sublinks of homology boundary links is not closed under positive mutation. We show that mutation does not preserve homology cobordism classes of 3-manifolds. A significant consequence is that there exist 3-manifolds which have the same quantum SU(2)-invariants but are not homology cobordant. These results are obtained by investigating the effect of mutation on the Milnor invariants, or equivalently the Massey products.
We prove that a fundamental group of codimension one nonnegative Ricci curvature C2-foliation of a closed Riemannian manifold is finitely generated and almost abelian, i.e. it contains abelian subgroup of finite index. In particular, we confirm the Milnor conjecture for manifolds which are leaves of codimension one nonnegative Ricci curvature foliation of closed manifold.
The Milnor degree of a 3-manifold is an invariant that records the maximum simplicity, in terms of higher order linking, of any link in the 3-sphere that can be surgered to give the manifold. This invariant is investigated in the context of torsion linking forms, nilpotent quotients of the fundamental group, Massey products and quantum invariants, and the existence of 3-manifolds with any prescribed Milnor degree and first Betti number is established. Along the way, it is shown that the number M(k,r) of linearly independent Milnor invariants of degree k, distinguishing r-component links in the…
We survey developments arising from Milnor's 1958 paper, "On the existence of a connection with curvature zero" and his 1977 paper, "On fundamental groups of complete affinely flat manifolds".
This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to many other locally symmetric spaces they do admit flat vector bundle of the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles …
We give a simple axiomatic definition of a rational-valued invariant s(W,V,e) of triples (W,V,e), where W is a (smooth, oriented, closed) 6-manifold and V is a 3-submanifold of W, and where e is a second rational cohomology class of the complement of V satisfying a certain condition. The definition is stated in terms of cobordisms of such triples and the signature of 4-manifolds. When W = S^6 and V is a smoothly embedded 3-sphere, and when e/2 is the Poincare dual of a Seifert surface of V, the invariant coincides with -8 times Haefliger's embedding invariant of (S^6,V). Our definition recover…
We investigate the notion of complexity for finitely presented groups and the related notion of complexity for three-dimensional manifolds. We give two-sided estimates on the complexity of all the Milnor groups (the finite groups with free action on the three-sphere), as well as for all finite Abelian groups. The ideas developed in the process also allow to construct two-sided bounds for the values of the so-called T-invariant (introduced by Delzant) for the above groups, and to estimate from below the value of T-invariant for an arbitrary finitely presented group. Using the results of this pa…
This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov rho-invariants associated to amenable representations. We introduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraic approximation we call local hidden torsion. We construct infinitely many hyperbolic 3-manifolds which have local hidden torsion in the transfinite lower central subgroup. By realizing Cheeger-Gromov invariants over amenable groups, we show that our hyperbolic 3-manifolds are not pairwise homology cobordant, yet remain indistingui…
In this paper (S_n) is a sequence of surfaces immersed in a 4-manifold which converges to a branched surface S_0. Up to sign, μ^T_p (resp. μ^N_p) will denote the amount of curvature of the tangent bundles TS_n (resp. the normal bundles NS_n) which concentrates around a singular point p of S_0 when n goes to infinity. By a slight abuse of notation, we call μ_p^T (resp. μ_p^N) the tangent (resp. normal) Milnor number of S_n at p. These numbers are not always well-defined; we discuss assumptions under which, if μ^T exists, then μ^N also exists and is smaller than -μ^T . When the second fundamenta…
In this paper we extend and Poincare dualize the concept of Euler structures, introduced by Turaev for manifolds with vanishing Euler-Poincare characteristic, to arbitrary manifolds. We use the Poincare dual concept, co-Euler structures, to remove all geometric ambiguities from the Ray-Singer torsion by providing a slightly modified object which is a topological invariant. We show that when the co-Euler structure is integral then the modified Ray-Singer torsion when regarded as a function on the variety of generically acyclic complex representations of the fundamental group of the manifold is …
We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. support norm) of the canonical contact structure, for each member of this family.
We derive formulae which lend themselves to TQFT interpretations of the Milnor torsion, the Lescop invariant, the Casson invariant, and the Casson-Morita cocyle of a 3-manifold, and, furthermore, relate them to the Reshetikhin-Turaev theory.
Milnor-Thurston homology theory is a construction of homology theory that is based on measures. It is known that it is equivalent to singular homology theory in case of manifolds and complexes. Its behaviour for non-tame spaces is still unknown. This paper provides results in this direction. We prove that Milnor-Thurston homology groups for the Warsaw Circle are trivial except for the zeroth homology group which is uncountable dimensional. Additionally, we prove that the zeroth homology group is non-Hausdorff for this space with respect a natural topology that was proposed by Berlanga.
We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tight contact structures that are not deformations of taut (or Reebless) foliations. This answers two questions of Etnyre.
In this paper, we exploit a subtle indeterminacy in the definition of the spherical Kervaire-Milnor invariant which was discovered by R. Stong to construct non-spin 4-manifolds with even intersection form and prescribed signature.
We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there is a degree-one map f:M --> M throwing K onto J and mapping M-K to M-J. These invariants are invariants of concordance and of J-characteristicness when f induces the identity on the fundamental group of M, and may be viewed as extensions of Milnor's invariants. We do not require the knots considered here to be rationally null-homologous or framed.
We establish several new results about both the (n)-solvable filtration, F_n^m, of the set of link concordance classes and the (n)-solvable filtration of the string link concordance group. We first establish a relationship between Milnor's invariants and links, L, with certain restrictions on the 4-manifold bounded by M_L. Using this relationship, we can relate (n)-solvability of a link (or string link) with its Milnor's invariants. Specifically, we show that if a link is (n)-solvable, then its Milnor's invariants vanish for lengths up to 2^{n+2}-1. Previously, there were no known results abou…
The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.
The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions.
We use a new geometric construction, grope splitting, to give a sharp bound for separation of surfaces in 4-manifolds. We also describe applications of this technique in link-homotopy theory, and to the problem of locating pi_1-null surfaces in 4-manifolds. In our applications to link-homotopy, grope splitting serves as a geometric substitute for the Milnor group.
Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\times S^2$. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orie…
A link in the 3-sphere is homotopically trivial, according to Milnor, if its components bound disjoint maps of disks in the 4-ball. This paper concerns the question of what spaces give rise to the same class of homotopically trivial links when used in place of disks in an analogous definition. We show that there are 4-manifolds for which this property depends on their embedding in the 4-ball. This work is motivated by the A-B slice problem, a reformulation of the 4-dimensional topological surgery conjecture. As a corollary this provides a new, secondary, obstruction in the A-B slice problem fo…
The notion of a Bing cell is introduced, and it is used to define invariants, link groups, of 4-manifolds. Bing cells combine some features of both surfaces and 4-dimensional handlebodies, and the link group λ(M) measures certain aspects of the handle structure of a 4-manifold M. This group is a quotient of the fundamental group, and examples of manifolds are given with π_1(M) not equal to λ(M). The main construction of the paper is a generalization of the Milnor group, which is used to formulate an obstruction to embeddability of Bing cells into 4-space. Applications to the A-B slice problem …
In this article, we study the topology of the family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ given by $F(x,y,z)=\bar{xy}(x^p+y^q)+z^r$ with $p,q,r \in \mathbb{N}$, $p,q,r \geq 2$ and $(p,q)=1$. Such a germ has isolated singularity at 0 and gives rise to a Milnor fibration $\frac{F}{|F|} \colon \mathbb{S}^{5} \setminus L_F \to \mathbb{S}^{1}$. We describe the link $L_F$ as a Seifert manifold and we show that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases the open-book decomposition of $\mathbb{S}^{5}$ …
The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4-manifold, generalizing a theorem of Milnor. Using this result, we give new obstructions to 3-manifolds being homology cobordant and to links being concordant. These obstructions are sufficiently strong to detect that the Bing double of the figure eight knot is not slice.
This paper makes certain observations regarding some conjectures of Milnor and Ramakrishnan in hyperbolic geometry and algebraic K-theory. As a consequence of our observations, we obtain new results and conjectures regarding the rationality and irrationality of Chern-Simons invariants of hyperbolic 3-manifolds.
This is the beginning of an obstruction theory for deciding whether a map f:S^2 --> X^4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall's self-intersection number mu(f) which tells the whole story in higher dimensions. Our second order obstruction tau(f) is defined if mu(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of pi_1(X) modulo S_3-symmetry (rather then just one copy modulo S_3-symmetry). It generalizes to the non…
An obstruction theory for representing homotopy classes of surfaces in 4-manifolds by immersions with pairwise disjoint images is developed, using the theory of non-repeating Whitney towers. The accompanying higher-order intersection invariants provide a geometric generalization of Milnor's link-homotopy invariants, and can give the complete obstruction to pulling apart 2-spheres in certain families of 4-manifolds. It is also shown that in an arbitrary simply connected 4-manifold any number of parallel copies of an immersed surface with vanishing self-intersection number can be pulled apart, a…
We consider Donaldson-Witten theory on four-manifolds of the form $X=Y \times {\bf S}^1$ where $Y$ is a compact three-manifold. We show that there are interesting relations between the four-dimensional Donaldson invariants of $X$ and certain topological invariants of $Y$. In particular, we reinterpret a result of Meng-Taubes relating the Seiberg-Witten invariants to Reidemeister-Milnor torsion. If $b_1(Y)>1$ we show that the partition function reduces to the Casson-Walker-Lescop invariant of $Y$, as expected on formal grounds. In the case $b_1(Y)=1$ there is a correction. Consequently, in the …
We introduce a homology surgery problem in dimension 3 which has the property that the vanishing of its algebraic obstruction leads to a canonical class of π-algebraically-split links in 3-manifolds with fundamental group π. Using this class of links, we define a theory of finite type invariants of 3-manifolds in such a way that invariants of degree 0 are precisely those of conventional algebraic topology and surgery theory. When finite type invariants are reformulated in terms of clovers, we deduce upper bounds for the number of invariants in terms of π-decorated trivalent graphs. We also con…
A shadowed polyhedron is a simple polyhedron equipped with half integers on regions, called gleams, which represents a compact, oriented, smooth 4-manifold. The polyhedron is embedded in the 4-manifold and it is called a shadow of that manifold. A subpolyhedron of a shadow represents a possibly singular subsurface in the 4-manifold. In this paper, we focus on contractible shadows obtained from the unit disk by attaching annuli along generically immersed closed curves on the disk. In this case, the 4-manifold is always a 4-ball. Milnor fibers of plane curve singularities and complexified real l…
The goal of this paper is to give a new proof of a theorem of Meng and Taubes that identifies the Seiberg-Witten invariants of 3-manifolds with Milnor torsion. The point of view here will be that of topological quantum field theory. In particular, we relate the Seiberg-Witten equations on a 3-manifold with the Abelian vortex equations on a Riemann surface. These techniques also give a new proof of the surgery formula for the Casson invariant, interpreted as an invariant of a homology S^2 x S^1.
Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are always quasi-isometric, and a group G is quasi-isometrically rigid if every group quasi-isometric to G is virtually isomorphic to G. In this survey we describe quasi-isometric rigidity results for fundamental groups of manifolds which can be decomposed into geometric pieces. After stating by now classical results on lattices in semisimple Lie groups, we focus …
Given a link L in the 3-sphere, we ask whether the components of L bound disjoint, nullhomologous disks properly embedded in a simply-connected positive-definite smooth 4-manifold; the knot case has been studied extensively in work of Cochran-Harvey-Horn. Such a 4-manifold is necessarily homeomorphic to a (punctured) connected sum of CP(2)'s. We characterize all links that are slice in a (punctured) connected sum of CP(2)'s in terms of ribbon moves and an operation which we call adding a generalized positive crossing. We find obstructions in the form of the Levine-Tristram signature function, …
We establish inequalities that constrain the genera of smooth cobordisms between knots in 4-dimensional cobordisms. These "relative adjunction inequalities" improve the adjunction inequalities for closed surfaces which have been instrumental in many topological applications of gauge theory. The relative inequalities refine the latter by incorporating numerical invariants of knots in the boundary associated to Heegaard Floer homology classes determined by the 4-manifold. As a corollary, we produce a host of concordance invariants for knots in a general 3-manifold, one such invariant for every n…
This paper concerns twisted signature invariants of knots and 3-manifolds. In the fibered case, we reduce the computation of these invariants to the study of the intersection form and monodromy on the twisted homology of the fiber surface. Along the way, we use rings of power series to obtain new interpretations of the twisted Milnor pairing introduced by Kirk and Livingston. This allows us to relate these pairings to twisted Blanchfield pairings. Finally, we study the resulting signature invariants, all of which are twisted generalisations of the Levine-Tristram signature.
This article surveys the use of configuration space integrals in the study of the topology of knot and link spaces. The main focus is the exposition of how these integrals produce finite type invariants of classical knots and links. More generally, we also explain the construction of a chain map, given by configuration space integrals, between a certain diagram complex and the deRham complex of the space of knots in dimension four or more. A generalization to spaces of links, homotopy links, and braids is also treated, as are connections to Milnor invariants, manifold calculus of functors, and…
