arXiv subjects

Mathieu Baillif

Publications and source records attributed to Mathieu Baillif.

21 records · Page 2Linked to original sources

arXiv2006

Homotopy in non metrizable omega-bounded surfaces

We investigate the problem of describing the homotopy classes $[X,Y]$ of continuous functions between $ω$-bounded non metrizable manifolds $X,Y$. We define a family of surfaces $X$ built with the first octant $C$ in $L^2$ ($L$ is the longline and $R$ the longray), and show that $[X,R]$ is in bijection with so called `adapted' subsets of a partially ordered set. We also show that $[M,R]$ can be computed for some surfaces $M$ that, unlike $C$, do not contain $R$. This indicates that when $X,Y$ are $ω$-bounded non metrizable surfaces, there might be a link between $[X,Y]$ and the concept of $Y$-directions in X$. Many pictures are used and the proofs are quite detailed.

math.GT↗

arXiv2004

Long line knots

We study continuous embeddings of the long line L into L^n (n>1) up to ambient isotopy of L^n. We define the direction of an embedding and show that it is (almost) a complete invariant in the case n=2 for continuous embeddings, and in the case n>3 for differentiable ones. Finally, we prove that the classification of smooth embeddings L \to L^3 is equivalent to the classification of classical oriented knots.

math.GN↗

arXiv2004

The homotopy classes of continuous maps between some non-metrizable manifolds

Let R be Alexandroff's long ray. We prove that the homotopy classes of continuous maps R^n \to R are in bijection with the antichains of P({1,...,n}). The proof uses partition properties of continuous maps R^n \to R. We also provide a description of $[X,R]$ for some other non-metrizable manifolds X.

math.GN↗