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]
617 questions
128
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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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28
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Homology in a picture? (Is this picture just metaphorical, or a rigorous example that can be formalized?)
A post-doc colleague showed me this picture and said: going from the diagram No.2 to No.3 and to No.4 is taking the homology.
I did not quite understand this comment. For me, if I take simplicial homology as an example, homology is setting up a…
gwynneth-m.sc.
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15
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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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2 answers
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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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
- 867
11
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1 answer
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
- 319
11
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1 answer
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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10
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Relation between local homeomorphism and homological dimension
For a given topological space $X$ (one can assume a simplicial complex if required), define it's homological dimension $\operatorname{hdim}(X)$ as the largest integer $n$ such that $H_n(X,A)\ne 0$ for some closed subspace $A$ of $X$.
I have been…
Rainy
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9
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Proof of the general case of Feynman's integration trick
I want to show that
$$\frac{1}{\displaystyle\prod_{i=0}^{i=n}A_{i}}=n!\int\limits_{|\Delta^{n}|}\frac{\mathrm d\sigma}{\left( \displaystyle \sum\limits_i s_i A_i \right)^n}$$
where $\mathrm d\sigma$ is the Lebesgue measure on the standard…
dbrane
- 241
8
votes
1 answer
Right split exact sequence for a Kan fibration with fiber a group complex
I'm stuck on the proof of Lemma 23.4 from May's Simplicial Objects in Algebraic Topology (p.99) on a seemingly harmless (and so left to reader's proof) step. Some context:
given a Kan complex $K$ with only one vertex $\phi\in K_0$ we have just…
8
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3 answers
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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Euler characteristic and the Mertens function?
I am interested if there can be given any applications of this topology on prime powers and a sheaf on it, to number theoretic questions ( I am looking for known results in elementary number theory, which could be proven with this topology or sheaf…
mathoverflowUser
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7
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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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1 answer
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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