Spaces of Geodesic Triangulations of Surfaces

We give a short proof of the contractibility of the space of geodesic triangulations with fixed combinatorial type of a convex polygon in the Euclidean plane. Moreover, for any $$n>0$$ n > 0 , we show that there exists a space of geodesic triangulations of a polygon with a triangulation, whose n-th homotopy group is not trivial.

Morse Index Bound for Minimal Two Spheres

Given a closed manifold of dimension at least three, with non-trivial homotopy group $$\pi _3(M)$$ π 3 ( M ) and a generic metric, we prove that there is a finite collection of harmonic spheres with Morse index bounded by one, with sum of their energies realizing a geometric invariant width.

Embedding spheres in knot traces

The trace of the $n$ -framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $2$ -sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $3$ -dimensional knot invariants. For each $n$ , this provides conditions that imply a knot is topologically $n$ -shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

Topological terms of (2+1)d flag-manifold sigma models

We examine topological terms of (2 + 1)d sigma models and their consequences in the light of classifications of invertible quantum field theories utilizing bordism groups. In particular, we study the possible topological terms for the U(N)/U(1)N flag-manifold sigma model in detail. We argue that the Hopf-like term is absent, contrary to the expectation from a nontrivial homotopy group π3(U(N)/U(1)N) = ℤ, and thus skyrmions cannot become anyons with arbitrary statistics. Instead, we find that there exist $$ \frac{N\left(N-1\right)}{2}-1 $$ N N − 1 2 − 1 types of Chern-Simons terms, some of which can turn skyrmions into fermions, and we write down explicit forms of effective Lagrangians.

On first countable quasitopological homotopy groups

If q: X → Y is a quotient map, then, in general, q × q: X × X → Y × Y may fail to be a quotient map. In this paper, by reviewing the concept of homotopy groups and quotient maps, we find under which conditions the map q × q is a quotient map, where q: Ω n (X, x 0) → πn (X, x 0), is the natural quotient map from the nth loop space of (X, x 0), Ω n (X, x 0), with compact-open topology to the quasitopological nth homotopy group πn (X, x 0). Ultimately, using these results, we found some properties of first countable homotopy groups.

Brunnian braids over the 2-sphere and Artin combed form

Finding homotopy group of spheres is an old open problem in topology. Berrick et al. derive in [A. J. Berrick, F. Cohen, Y. L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006)] an exact sequence that relates Brunnian braids to homotopy groups of spheres. We give an interpretation of this exact sequence based on the combed form for braids over the sphere developed in [R. Gillette and J. V. Buskirk, The word problem and consequences for the braid groups and mapping class groups of the two-sphere, Trans. Amer. Math. Soc. 131 (1968) 277–296] with the aim of helping one to visualize the sequence and to do calculations based on it.

Some results in quasitopological homotopy groups

UDC 515.4 We show that the th quasitopological homotopy group of a topological space is isomorphic to th quasitopological homotopy group of its loop space and by this fact we obtain some results about quasitopological homotopy groups. Finally, using the long exact sequence of a based pair and a fibration in qTop introduced by Brazas in 2013, we obtain some results in this field.

The groups (2, 𝑚 | 𝑛, 𝑘 | 1, 𝑞): Finiteness and homotopy

We initiate the study of the groups (l,m\mid n,k\mid p,q) defined by the presentation \langle a,b\mid a^{l},b^{m},(ab)^{n},(a^{p}b^{q})^{k}\rangle. When p=1 and q=m-1, we obtain the group (l,m\mid n,k), first systematically studied by Coxeter in 1939. In this paper, we restrict ourselves to the case l=2 and \frac{1}{n}+\frac{1}{k}\leq\frac{1}{2} and give a complete determination as to which of the resulting groups are finite. We also, under certain broadly defined conditions, calculate generating sets for the second homotopy group \pi_{2}(Z), where 𝑍 is the space formed by attaching 2-cells corresponding to (ab)^{n} and (ab^{q})^{k} to the wedge sum of the Eilenberg–MacLane spaces 𝑋 and 𝑌, where \pi_{1}(X)\cong C_{2} and \pi_{1}(Y)\cong C_{m}; in particular, \pi_{1}(Z)\cong(2,m\mid n,k\mid 1,q).

On the non-abelian tensor square of all groups of order dividing p5

In this paper, we consider all groups of order dividing [Formula: see text]. We obtain the explicit structure of the non-abelian tensor square, non-abelian exterior square, tensor center, exterior center, the third homotopy group of suspension of an Eilenberg–MacLane space [Formula: see text] and [Formula: see text] of such groups.