In [3] Mather proved that a smooth proper infinitesimally stable map is stable. This result is the key component of the Mather stability theorem [5], which can be reformulated as follows: a smooth proper map f : M -> Nis stable if and only if it is infinitesimally stable if and only if it satisfies the Mather normal crossing condition. The latter condition, roughly speaking, means that all map germs of f are stable and f maps the singular strata of f to Nin a mutually transversal manner. In this note we adapt a short argument from the book [2] to derive the Mather stability theorem presented in [5] from the theorem in [3]. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Morse functions on a closed manifold M need not realize the minimal number of critical points of smooth functions on M since several critical points often may be fused into a single degenerate one. We study conditions under which critical points of smooth functions on a manifold M of dimension 3 can be fused. We say that a function f:M→ [a, c] on a compact manifold is regular if the boundary of M is the union of f^-1(a) and f^-1(c) , and f has no critical points on ∂ M . We prove that if f^-1(a) is a sphere and f^-1(c) is a non-empty surface, then the Conley index is a complete obstruction to fusing critical points of a regular function. On the other hand, we prove that the Conley index is not a complete obstruction to fusing critical points if, for example, M is F_0,3× S^1 , where F_0,3 is a 2-sphere with the interior of three disjoint disks removed.

Kauffman virtual knots are knots in thickened surfaces F×R considered up to isotopy, stabilizations and destabilizations, and diffeomorphisms of F×R induced by orientation preserving diffeomorphisms of F. Similarly, virtual Legendrian knots, introduced by Cahn and Levi, are Legendrian knots in ST∗F with the natural contact structure. Virtual Legendrian knots are considered up to isotopy, stabilization and destabilization of the surface away from the front projection of the Legendrian knot, as well as up to contact isomorphisms of ST∗F induced by orientation preserving diffeomorphisms of F. We show that there is a projection operation proj from the set of virtual isotopy classes of Legendrian knots to the set of isotopy classes of Legendrian knots in ST∗S2. This projection is obtained by substituting some of the classical crossings of the front diagram with virtual crossings. It restricts to the identity map on the set of virtual isotopy classes of classical Legendrian knots. In particular, proj extends invariants of Legendrian knots to invariants of virtual Legendrian knots. Using proj, we show that the virtual crossing number of every classical Legendrian knot equals its crossing number. We also prove that the virtual canonical genus of a Legendrian knot is equal to the canonical genus. The construction of proj is inspired by the work of Manturov.

. The singular set of a generic map from a closed manifold of dimension at least 2 to the plane is a smooth closed curve. We study the parity of the number of components of the singular set under the assumption that the map is an image simple fold map, i.e., the map's restriction to its singular set is a smooth embedding. The image of the singular set of a map to a plane inherits canonical local orientations via so-called chessboard functions. Such a local orientation gives rise to a cumulative winding number, which is an integer or a half integer. When the dimension of the source manifold is even, we also define an invariant I which is the residue class modulo 4 of the sum of twice the number of components of the singular set, the number of cusps, and twice the number of self-intersection points of the image of the singular set. Using the cumulative winding number and the invariant I, we show that the parity of the number of connected components of the singular set does not change under homotopy between image simple fold maps provided that one of the following conditions is satisfied: (i) the dimension of the source manifold is even, (ii) the image of the singular set of the homotopy does not have triple self-intersection points, or (iii) the singular set of the homotopy is an orientable manifold with boundary.

Introduced by Seifert and Threlfall, cylindrical neighborhoods of isolated critical points of smooth functions is an essential tool in the Lusternik- Schnirelmann theory. We conjecture that every isolated critical point of a smooth function admits a cylindrical ball neighborhood. We show that the conjecture is true for cone-like critical points, Cornea reasonable critical points, and critical points that satisfy the Rothe H hypothesis. In particular, the conjecture holds true at least for those critical points that are not infinitely degenerate. If, contrary to the assertion of the conjecture, there are isolated critical points that do not admit cylindrical ball neighborhoods, then we say that such critical points are exotic. We prove a Lusternik-Schnirelmann type theorem asserting that the minimal number of critical points of smooth functions without exotic critical points on a closed manifold of dimension at least 6 is the same as the minimal number of elements in a Singhof-Takens filling of M by smooth balls with corners.

We formulate conjectures generalizing some known results to the category of virtual Legendrian knots. This includes statements relating virtual Legendrian knots to ordinary Legendrian knots, non-existence of positive virtual Legendrian self isotopy for the class of the fiber of $ST^*M$ and the conjectural relation of virtual Legendrian isotopy to causality in generalized spacetimes. We prove the conjectures in the case of $2$-dimensional $M$ and $(2+1)$-dimensional spacetimes. We also formulate and prove the version of the Arnold's $4$ cusp conjecture for virtual isotopies.

The singular set of a generic map $f: M\to F$ of a manifold $M$ of dimension $m\ge 2$ to an oriented surface $F$ is a closed smooth curve $\Sigma(f)$. We study the parity of the number of components of $\Sigma(f)$. The image $f(\Sigma)$ of the singular set inherits canonical local orientations via so-called chessboard functions. Such a local orientation gives rise to the cumulative winding number $\omega(f)\in \frac{1}{2}\mathbb{Z}$ of $\Sigma(f)$. When the dimension of the manifold $M$ is even we also define an invariant $I(f)$ which is the residue class modulo $4$ of the sum of the number of components of $\Sigma(f)$, the number of cusps, and twice the number of self-intersection points of $f(\Sigma)$. Using the cumulative winding number and the invariant $I(f)$, we show that the parity of the number of connected components of $\Sigma(f)$ does not change under homotopy of $f$ provided that one of the following conditions is satisfied: (i) the dimension of $M$ is even, (ii) the singular set of the homotopy is an orientable manifold, or (iii) the image of the singular set of the homotopy does not have triple self-intersection points.

Introduced by Seifert and Threlfall, cylindrical neighborhoods is an essential tool in the Lusternik-Schnirelmann theory. We conjecture that every isolated critical point of a smooth function admits a cylindrical ball neighborhood. We show that the conjecture is true for cone-like critical points, Cornea reasonable critical points, and critical points that satisfy the Rothe H hypothesis. In particular, the conjecture holds true at least for those critical points that are not infinitely degenerate.

We find conditions under which a non-orientable closed surface S embedded into an orientable closed 4-manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4-sphere. This allows us to extend the Gabai 4-dimensional light bulb theorem and the Auckly-Kim-Melvin-Ruberman-Schwartz one is enough theorem to the case of non-orientable surfaces.

A flat virtual link is a finite collection of oriented closed curves [Formula: see text] on an oriented surface [Formula: see text] considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves [Formula: see text], we show that the minimal number of intersection points of curves in the virtual homotopy class of [Formula: see text] equals to the number of terms of a generalization of the Anderson–Mattes–Reshetikhin Poisson bracket. Furthermore, considering a single curve, we show that the minimal number of self-intersections of a curve in its virtual homotopy class can be counted by a generalization of the Cahn cobracket.

We use the Berstein-Hilton invariant to prove the formula $\cat(M_1\sharp M_2)=\max\{\cat M_1, \cat M_2\}$ for the Lustrnik-Schnirelmann category of the connected sum of closed manifolds $M_1$ and $M_2$.

We prove the formula $TC(G\ast H)=\max\{TC(G), TC(H), cd(G\times H)\}$ for the topological complexity of the free product of discrete groups with cohomological dimension >2.

The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\cat(M\# N)=\max\{\cat M,\cat N\}$. For topological complexity we prove the inequality $\TC (M\# N)\ge\max\{\TC M,\TC N\}$ for simply connected manifolds.

We give elementary proofs of the Takeuchi and He theorems on the real cohomology rings and real equivariant cohomology rings of real Grassmann manifolds.

In this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the b-principle and in dimension 2 approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension 2 to a closed simply connected manifold is bordant to a submersion. We show that the Madsen-Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b-principle. Namely, a version of the b-principle for oriented colored broken submersions together with the Harer stability theorem and Miller-Morita theorem implies the Madsen-Weiss theorem.

An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. In contrast to Legendrian knots, virtual Legendrian knots enjoy the property that there is a bijective correspondence between the virtual Legendrian knots and the equivalence classes of Gauss diagrams. We study virtual Legendrian isotopy classes of Legendrian links and show that every such class contains a unique irreducible representative. In particular we get a solution to the following conjecture of Cahn, Levi and the first author: two Legendrian knots in $ST^*S^2$ that are isotopic as virtual Legendrian knots must be Legendrian isotopic in $ST^*S^2.$

We classify capacities on the class Sym o (2) of connected symplectic surfaces with at most countably many nonplanar ends. To obtain the classification we study diffeomorphism types of surfaces in Sym o (2) of infinite genus with nonplanar ends; it turns out that these types are in bijective correspondence with countable successor ordinals of the form ω α · d + 1, where α is an ordinal and d ≥ 0 is an integer. It also turns out that if S 1 and S 2 are two open surfaces of infinite genera with at most countably many nonplanar ends, then each of the surfaces embeds into the other. Our classification implies that every capacity on the class of symplectic surfaces in Sym o (2) of infinite genus differs from the Hofer–Zehnder capacity by a non-negative finite or infinite constant.

We review the bordism version of the h-principle established in an earlier work by the author and discuss its generalization.

In view of the self-linking invariant, the number $|K|$ of framed knots in $S^3$ with given underlying knot $K$ is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that $|K|$ is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of $S^1\times S^2$; the knot $K$ need not be zero-homologous and the manifold is not required to be compact. We show that when $M$ is orientable, the number $|K|$ is infinite unless $K$ intersects a non-separating sphere at exactly one point, in which case $|K|=2$; the existence of a non-separating sphere implies that $M$ contains a connected sum factor of $S^1\times S^2$. For knots in nonorientable manifolds we show that if $|K|$ is finite, then $K$ is disorienting, or there is an isotopy from the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded $S^2$ or $\mathbb R P^2$ at exactly one point, or it intersects some embedded $S^2$ at exactly two points in such a way that a closed curve consisting of an arc in $K$ between the intersection points and an arc in $S^2$ is disorienting.

We show that for any n real periodic functions f_1,..., f_n with the same period, such that f_i>0 for i 0, there is a closed curve in R^{n+1} with curvatures k_1, ..., k_n such that |k_i(t)-f_i(t)| < e for all i and t. This neither holds for closed curves in the hyperbolic space H^{n+1}, nor for parametric families of closed curves in R^{n+1}.