I know that a requirement for a finite connected simplicial complex $K$ to be a closed combinatorial surface is that the link of any vertex is a simplicial circle. So, suppose that $C$ is a subcomplex of $K$ such that $C$ is a simplicial circle. Is it generally the case that there is a vertex $v \in K$ such that the link of $v$ is $C$?
2026-03-26 07:30:56.1774510256
Subcomplexes of a Closed Combinatorial Surface
101 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GENERAL-TOPOLOGY
- Is every non-locally compact metric space totally disconnected?
- Let X be a topological space and let A be a subset of X
- Continuity, preimage of an open set of $\mathbb R^2$
- Question on minimizing the infimum distance of a point from a non compact set
- Is hedgehog of countable spininess separable space?
- Nonclosed set in $ \mathbb{R}^2 $
- I cannot understand that $\mathfrak{O} := \{\{\}, \{1\}, \{1, 2\}, \{3\}, \{1, 3\}, \{1, 2, 3\}\}$ is a topology on the set $\{1, 2, 3\}$.
- If for every continuous function $\phi$, the function $\phi \circ f$ is continuous, then $f$ is continuous.
- Defining a homotopy on an annulus
- Triangle inequality for metric space where the metric is angles between vectors
Related Questions in ALGEBRAIC-TOPOLOGY
- How to compute homology group of $S^1 \times S^n$
- the degree of a map from $S^2$ to $S^2$
- Show $f$ and $g$ are both homeomorphism mapping of $T^2$ but $f$ is not homotopy equivalent with $g.$
- Chain homotopy on linear chains: confusion from Hatcher's book
- Compute Thom and Euler class
- Are these cycles boundaries?
- a problem related with path lifting property
- Bott and Tu exercise 6.5 - Reducing the structure group of a vector bundle to $O(n)$
- Cohomology groups of a torus minus a finite number of disjoint open disks
- CW-structure on $S^n$ and orientations
Related Questions in SIMPLICIAL-COMPLEX
- What is the intersection of the vertices of a face of a simplicial complex?
- Simplicial complex underlying space homeomorphic to torus/klein bottle
- Fixed points of simplicial maps
- Homotopy type of simplicial complexes
- Non-contractible simplicial complex and Euler characteristic 1
- Persistence Homology on a grid Distance measure
- Finding chains with a given boundary
- Is there a time where the link of a subset is not equal to the boundary of its star?
- Edge-to-edge incidence structure of a graph
- Triangulation of the projective plane (number of points required)
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
To the contrary, it is not generally the case.
If $K$ is not homeomorphic to a sphere then it is never the case, because $K$ always has a simplicial circle $C$ which is not homotopic to a constant, namely the image of the shortest closed edge path which is not path homotopic to a constant.
And if $K$ is homeomorphic to a sphere then it might be true, but still I would say that the simplicial structures on a sphere which have that property are an exception: imagine a very fine simplicial structure on the sphere for which the equator is a simplicial circle.