topological manifold
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Author(s):  
Arturo Tozzi

When an edge is removed, a cycle graph Cn becomes a n-1 tree graph. This observation from extremal set theory leads us to the realm of set theory, in which a topological manifold of genus-1 turns out to be of genus-0. Starting from these premises, we prove a theorem suggesting that a manifold with disjoint points must be of genus-0, while a manifold of genus-1 cannot encompass disjoint points.



2021 ◽  
Author(s):  
Yu-Lin Chou

We show as a main message that there is a simple dimension-preserving way to openly and densely embed every topological manifold into a compact ``nonstandard'' topological manifold with boundary.This class of ``nonstandard'' topological manifolds with boundary contains the usual topological manifolds with boundary.In particular,the Alexandroff one-point compactification of every given topological $n$-manifold is a ``nonstandard'' topological $n$-manifold with boundary.



2020 ◽  
Vol 148 (9) ◽  
pp. 3787-3799
Author(s):  
George Raptis ◽  
Wolfgang Steimle






2018 ◽  
Vol 08 (04) ◽  
pp. 134-142
Author(s):  
Jean-Francois Niglio


2017 ◽  
Vol 26 (02) ◽  
pp. 1740005 ◽  
Author(s):  
Robert E. Gompf

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least two arise in this manner. Most Euclidean spaces of dimensions five and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For [Formula: see text], there is a topological flow on (ℝ2r+1 − 8 points) × ℝm that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.



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