Given a virtual link diagram *D*, we define its unknotting index to be minimum among tuples, where *m* stands for the number of crossings virtualized and *n* stands for the number of classical crossing changes, to obtain a trivial link diagram. By using span of a diagram and linking number of a diagram we provide a lower bound for unknotting index of a virtual link. Then using warping degree of a diagram, we obtain an upper bound. Both these bounds are applied to find unknotting index for virtual links obtained from pretzel links by virtualizing some crossings.

We prove a selection theorem for convex-valued lower semicontinuous mappings Fto Fréchet spaces under the assumption that every value of Fcontains all interior (in the convex sense) points of its own closure. This is an extension of E. Michael’s theorem 3.1''' in [7]for mappings to Banach spaces. The desired continuous single-valued selection is constructed as a pointwise barycenter mapping with respect to a suitable family of probability measures concentrated on values of F. As an application, we show that, for any metric space Mthere is a continuous mapping which to every compact set K⊂Massigns a probability measure whose support coincides with K. Earlier this fact was proved for a complete separable metric space Mby using a fundamentally different technique based on Milyutin mappings.

In this paper we provide some affirmative results and some counterexamples for a solution of the splitting problem for n multivalued mappings, n>2.

We prove that there is an algorithm which determines whether or not a given 2-polyhedron can be embedded into some integral homology 3-sphere. This is a corollary of the following main result. Let M be a compact connected orientable 3-manifold with boundary. Denote G=Z, G=Z/pZ or G=Q. If H1(M;G)≅Gk and ∂M is a surface of genus g, then the minimal group H1(Q;G) for closed 3-manifolds Q containing M is isomorphic to Gk−g. Another corollary is that for a graph L the minimal number.

If *g *is a map from a space *X *into R*m *and *q *is an integer, let *B q**,**d**,**m**(**g**) *be the set of all planes *Π**d* ⊂R*m *such that |*g*−1*(**Π**d**)*| ≥ *q*. Let also H*(**q**,**d**,**m**,**k**) *denote the set of all continuous maps *g *: *X *→R*m *such that dim *B q**,**d**,**m**(**g**) *≤*k*. We prove that for any *n*-dimensional metric compactum *X *each of the sets H*(*3*, *1*,**m**, *3*n *+ 1 −*m**) *and H*(*2*, *1*,**m**, *2*n**) *is dense and *G**δ* in the function space *C**(**X**,* R*m**) *provided *m *≥ 2*n *+ 1 (in this case H*(*3*, *1*,**m**, *3*n *+ 1 − *m**) *and H*(*2*, *1*,**m**, *2*n**) *can consist of embeddings). The same is true for the sets H*(*1*,**d**,**m**,**n*+ *d**(**m *− *d**)) *⊂ *C**(**X**,* R*m**) *if *m *≥*n *+ *d*, and H*(*4*, *1*, *3*, *0*) *⊂ *C**(**X**,* R*3**) *if dim *X *≤ 1.

We construct combinatorial formulas of Fiedler type (i.e. composed of oriented Gauss diagrams arranged by homotopy classes of loops in the base manifold, see [T. Fiedler, Gauss Diagram Invariants for Knots and Links, Math. Appl., vol. 552, Kluwer Academic Publishers, 2001; M. Polyak, O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not. 11 (1994) 445–453]) for an infinite family of finite type invariants of knots in M^2 х R^1 (M^2 orientable), introduced in [S.A. Grishanov, V.A. Vassiliev, Two constructions of weight systems for invariants of knots in non-trivial 3-manifolds, Topology Appl. 155 (2008) 1757–1765]

In the standard approach to pointfree topology via frames/locales, a topological space is represented by its frame of open subsets. The aim of this note is to propose an alternative approach, by representing a topological space by a frame of continuous functions with values in what we call a *topological frame*. A prime example of such a familiar representation is the frame of lower semicontinuous [0, \infty]-valued functions on a topological space.

Let MS^{t}(M^n,k)$ and $MS(M^n,k)$ be Morse-Smale flows and diffeomorphisms respectively the non-wandering set of those consists of $k$ fixed points on a closed $n$-manifold $M^n$. For $k=3$, we show that the only values of $n$ possible are $n\in\{2,4,8,16\}$, and $M^2$ is the projective plane. For $n\geq 4$, $M^n$ is simply connected and orientable. We prove that the closure of any separatrix of $f^t\in MS^{t}(M^n,3)$ is a locally flat n/2-sphere while there is $f^t\in MS^{t}(M^n,4)$ such that the closure of separatrix of $f^t$ is a wildly embedded codimension two sphere. This allows us to classify flows from $MS^{t}(M^4,3)$. For n>5, one proves that the closure of any separatrix of $f\in MS(M^n,3)$ is a locally flat n/2-sphere while there is $f\in MS(M^4,3)$ such that the closure of any separatrix is a wildly embedded 2-sphere.

In the present paper we prove that frames of one-dimensional separatrices in basins of sources of structurally stable 3-diffeomorhisms with two-dimensional expanding attractor are trivially embedded. This result plays an important part in the classification of such systems. The classification was given by V. Grines and E. Zhuzhoma with assumption that all one-dimensional separatrices are trivially embedded into the ambient manifold but the proof of the assumption was never given. Thus, the present paper is a necessary and nontrivial element of the classification of structurally stable diffeomorphisms with codimension one expanding attractors.

In 1965, P.F. Baum and W. Browder proved that RP10 cannot be immersed to R15. Going alternative way, we investigate this problem using U. Koschorke’ singularity approach. In this paper, we simplify and analyze the corresponding obstruction group.

A new selection theorem is proved for lower semicontinuous mappings F:X→LM(T;B)into an Orlisz spaces of summable mappings. Values F(x) of *F* in this theorem in general are neither convex nor decomposable, they are unions of two sets which are both convex and decomposable. The key ingredient of the proof is an appropriate estimate of nonconvexity of such union.

For the Jennings group J(k) of substitutions of formal power series with coefficients in a field k of positive characteristic (the Nottingham group), the depth of the powers of its elements is studied. In particular, it is shown that the case of a field with characteristic 2 is completely different from the case of a field with odd prime characteristic. It is also shown that the case of the field k=Z_{2 differs from the case of other fields with characteristic 2. Explicit embeddings of the groups Zpm and Zp⊕Zp in J(Zp) are constructed.}

We find the high commutants of the Jennigs group J(Z_2) of substitutions of formal power series with coefficients in the ring Z_2. We explicitly provide the corresponding abelianizing homomorphisms.

A new family of weight systems of finite type knot invariants of any positive degree in orientable 3-manifolds with non-trivial first homology group is constructed. The principal part of the Casson invariant of knots in such manifolds is split into the sum of infinitely many independent weight systems. Examples of knots separated by corresponding invariants and not separated by any other known finite type invariants are presented.