A finite simplicial complex can be defined as a finite collection $K$ of simplices in $\mathbb{R}^N$ that satisfies the following conditions : (1) Any face of a simplex from $K$ is also in $K$ and (2) The intersection of any two simplices in $K$ is a face of both simplices.

# Questions tagged [simplicial-complex]

471 questions

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### Simplicial Complex vs Delta Complex vs CW Complex

I am a little confused about what exactly are the difference(s) between simplicial complex, $\Delta$-complex, and CW Complex.
What I roughly understand is that $\Delta$-complexes are generalisation of simplicial complexes (without the requirement…

yoyostein

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### Has anyone ever actually seen this Daniel Biss paper?

A student asked me about a paper by Daniel Biss (MIT Ph.D. and Illinois state senator) proving that "circles are really just bloated triangles." The only published source I could find was the young adult novel An Abundance of Katherines by John…

nardol5

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### Show that the $\Delta$-complex obtained from $\Delta^3$ by performing edge identifications deformation retracts onto a Klein bottle.

I am going through some exercises in Hatcher's Algebraic Topology.
You have a $\Delta$-complex obtained from $\Delta^3$ (a tetrahedron) and perform edge identifications $[v_0,v_1]\sim[v_1,v_3]$ and $[v_0,v_2]\sim[v_2,v_3]$. How can you show that…

09867

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### When is the quotient of simplicial complexes a simplicial complex?

Let $K$ be a simplicial complex and let $L$ be a subcomplex of $K$.
Questions:
Is it possible to define an operation on (some) simplicial complexes so that $K/L$ is a simplicial complex for which $|K/L|\cong |K|/|L|$?
Is it the case that $|K|/|L|$…

user12344567

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### Natural equivalence between singular and simplicial homology

I know that, for every $\Delta$-complex $X$, there is a canonical isomorphism $\phi_n : H_n ^\Delta (X) \to H_n (X) $, where $H^\Delta _n (X)$ is the $n$-th simplicial homology group, and $H_n (X)$ is the $n$-th singular homology group.
For…

Ervin

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### Retraction onto a circle in a simplicial complex

Let $X$ be a connected space homeomorphic to a finite simplicial complex. If there is an embedding $i: S^1 \hookrightarrow X$ which has a retract $r: X \rightarrow S^1$, then necessarily the first Betti number $b_1(X)$ is nonzero. Is the condition…

Cihan

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### Extensions of Sperner families on finite sets

Let $V$ be a set consisting of $n$ points, $n\geq 2$.
A Sperner family on $V$ is a set $\{\sigma_1,\sigma_2,\dots,\sigma_m\}$, $m\geq 1$, where each $\sigma_i$, $i=1,2,\dots,m $, is a nonempty subset of $V$, and for any $i\neq j$, $\sigma_i$ does…

CW Complex

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### So what is Cohomology?

I have few questions about Cohomology, all related to each other. Please assume I have minimal knowledge of the subject and I need to have even basic things explained.
1) What is Cohomology? On the most basic level, what do we try to achieve by it?…

tomers99

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### A topological example from Church's undecidability paper

A. Church, in his classical paper An unsolvable problem in elementary number theory in American Journal of Mathematics Vol. 58 No. 2. (1936), pp. 345-363, (available here), wrote:
There is a class of problems of elementary number
theory which…

Jacopo Notarstefano

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### Why does the Betti number give the measure of k-dimensional holes?

I was reading Paul Renteln "MANIFOLDS, TENSORS, AND FORMS An Introduction for Mathematicians and Physicists" p.145, where he defined the Betti number as $dim H_m(K)$, where $H_m(K)$ is the quotient space of cycles modulo boundary $Z_m(K)/B_m(K)$. He…

Joe Martin

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### Fixed points of simplicial maps

I want to prove a statement about the fixed points of simplicial maps.
If $f: |K|\to |K|$ is a simplicial map prove that the set of fixed points of $f$ is the polyhedron of a subcomplex of $K^1$ (where $K^1$ denotes the first barycentric…

Polymorph

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### How does one orient a simplicial complex?

I have a simplicial complex, built out of hyper-tetrahedra (5-cells) with the topology of $S_{4}$ and I would like to assign an ordering to it's vertices (some couple thousand), so that I can apply a boundary operator and co-boundary operator on…

kηives

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### Understanding the $\Delta$-complex structure of a quotient space

I'm reading Hatcher's book of Algebraic Topology where he define the notion of $\Delta$-complex. Then he puts the following diagram
And said that this shows a $\Delta$-complex structure of the torus, the real projective plane and the Klein bottle.…

Walter Simon

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### Why all the alternating signs?

For instance:
$ |A \cup B \cup C|=(|A|+|B|+|C|)-(|A \cap B| + |A \cap C| + |B \cap C|)+|A \cap B \cap C|$
$\chi(X) = F- E + V = \sum_{i} (-1)^i \text{rank}(H_i(X)) =\sum_{i} (-1)^i \text{rank}(C_i(X))$
Differential forms/Exterior algebra
$\partial…

Robin

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### Nontrivial cup product realized in $\Bbb R^4$

Let $A$ be a closed subspace $A$ of $[0,1]^4$---let's say, a subcomplex of some triangulation of the cube. I would like to show that the cup product $H^2(A)\times H^2(A)\to H^4(A)$ is trivial (or at least that the square map $x\mapsto x\smile x$ is…

Peter Franek

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