Skip to main content

Questions tagged [simplicial-complex]

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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
56 views

If $G := \langle a_k : k \in \mathbb{Z} \rangle$ and $H:= \langle a_{k+1} - a_k : k \in\mathbb{Z} \rangle$, Prove that $G/H \cong \mathbb{Z}$

I am trying to prove that if $G := \langle a_k : k \in \mathbb{Z} \rangle$ and $H:= \langle a_{k+1} - a_k : k \in\mathbb{Z} \rangle$ with both of them being free abelian groups, then $G/H \cong \...
Squirrel-Power's user avatar
0 votes
1 answer
41 views

How to understand May's proof that counit map is a weak equivalence?

A similar question was asked about 4 years ago here, but received no answers, so I hope it is appropriate to post a new question. I am trying to read the singular homology section in May's Concise ...
Christian's user avatar
0 votes
0 answers
27 views

Definition for orders corresponding to directed acyclic graphs (DAG)

My question What is the name for a binary relation $R$ on $V$ that corresponds to a graph $G = (V,E)$ that is a directed acyclic (simple) graph? Background There is a bijection between simple directed ...
Berber's user avatar
  • 394
1 vote
1 answer
74 views

Nerve theorem for small permutative categories

For small categories, there is a famous Nerve theorem: A simplicial set $X:\Delta^{op}\to Set$ is a nerve of a small category if and only if it satisfies the Segal conditions. For small permutative ...
xuexing lu's user avatar
2 votes
1 answer
39 views

Is a strict fundamental domain in a simply connected complex simply connected itself?

Let $G$ be a group acting on a simplicial (or just combinatorial CW-) complex by simplicial morphisms without inversion. Assume that this action has a strict fundamental domain $Y\subseteq X$, meaning ...
Giraud's user avatar
  • 75
2 votes
3 answers
63 views

Showing that the inclusion of the 2-skeleton of a simplicial complex induces an isomorphism on fundamental groups

Suppose that $K$ is a simplicial complex and let $K_{(2)}$ denote its 2-skeleton. Show that that for all $x \in |K_{(2)}|$ the map $i_*: \pi_1(|K_{(2)}|,x) \to \pi_2(|K|,x)$ induces an isomorphism of ...
John Robertson's user avatar
5 votes
0 answers
75 views

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 ...
Alberto Avitabile's user avatar
1 vote
0 answers
54 views

What on earth is a topological cavity in a certain simplicial complex? [closed]

I am dealing with basic algebraic topology for my research, and have been confused about topological cavities for a long time. Until now, I have already understood the concepts about k-dimensional ...
dhliu's user avatar
  • 11
0 votes
0 answers
106 views

Simplicial complex construction of the prefix set of tuples

For $(S_i)_{i \in I}$ a familiy of sets, the set of all prefixes of tuples in $\prod_{i \in I} S_i$ is $$ \prod_{i\in I} (S_i \dot{\cup} \{\bot\}) \;,$$ which is just a choice of either some element ...
Berber's user avatar
  • 394
1 vote
1 answer
45 views

Proving that the cone on a finite Euclidean simplicial complex is again a Euclidean simplicial complex.

Background This is Problem 5-17 of John Lee's Introduction to Topological Manifolds. Suppose $\sigma=[v_0,\ldots,v_k]$ is a simplex in $\mathbb{R} ^n$ and $w\in \mathbb{R} ^n$. If $\{w,v_0,\ldots,v_k\}...
Ricky's user avatar
  • 3,169
1 vote
1 answer
65 views

Algorithm for a simplicial presentation complex

Let $G$ be a finitely presented group with a finite presentation. Given this presentation, I want to construct via an algorithm a finite, $2$-dimensional simplicial complex $X$ with fundamental group $...
The_Rookie's user avatar
3 votes
1 answer
73 views

Steenrod squares and higher cup products for differential forms?

I am physicist, so I am sorry if I am not too rigorous in the following. I have two (closely related I guess) questions: Let me consider a triangulated manifold $M$ and its simplicial cohomology. ...
Weyl's user avatar
  • 31
1 vote
0 answers
50 views

Minimal Simplicial Complex from a Sequence of Betti Numbers

I found the following problem in a Computational Topology course that I am following: Write an algorithm that given a sequence $(\beta_0,\ldots,\beta_d)$ of integers builds a simplicial complex whose $...
Pepe's user avatar
  • 115
1 vote
0 answers
58 views

Algorithmic construction of a loop

Let $X$ be a finite simplicial complex such that its geometric realisation $|X|$ is homeomorphic to a closed, connected, orientable manifold $M^n$. Suppose we are given an element $\gamma \in \pi_1(|X|...
The_Rookie's user avatar
2 votes
0 answers
47 views

Is there an algorithm for finding a geometric realization of a finite abstract simplicial complex [closed]

I have a $d$-dimensional finite abstract simplicial complex (ASC); that is I have a sequence of incidence matrices that define the complex. I know that any such ASC can be embedded in ${\mathbb R}^N$ ...
unknown's user avatar
  • 1,010
1 vote
0 answers
41 views

Triangulating Product of Simplicial Complexes

I am currently working on a problem for which I believe the following result is crucial. The result of this problem was discussed in this post. Product of simplicial complexes? However it is not ...
slowlight's user avatar
  • 353
0 votes
0 answers
19 views

Automorphism group of a product of two simplices

Is there a description of the group of automorphisms of a product of two simplices depending on the dimension of the simplices? By automorphisms, I mean simplicial automorphisms of a simplicial ...
Grisha Taroyan's user avatar
0 votes
0 answers
22 views

Defining gluing data and orientation of simplices.

I am currently learning the basics of simplical homology and have been reading about gluing data and boundary maps. My question is when talking about choosing gluing data by defining map $\beta(\alpha,...
zak zaki's user avatar
  • 151
3 votes
1 answer
160 views

Singular Homology is a special case of Simplicial Homology

Hatcher's Algebraic Topology states on page 108 that singular homology is a special case of simplicial homology: Though singular homology looks so much more general than simplicial homology, it can ...
RyeCatcher's user avatar
0 votes
0 answers
118 views

Motivation about "Analysis Situs"

I read Poincaré's paper called Analysis Situs. And here's the thing about chain complex. (Page 104 in this file) That being given, let ${ε}^q_{i,j}$ be a number which is equal to zero if ${a}^{q−1}...
user1274233's user avatar
0 votes
1 answer
52 views

Homotopy equivalences, removing irrelevant subcomplex

With the following information: $X$ is a finite simplicial complex of dimension $n$. $W$ is a simplicial subcomplex of $X$ of dimension $d<n$. (Edit) All the simplices of dimension less or equal ...
MathBug's user avatar
  • 394
0 votes
1 answer
62 views

The formal definition of a Δ-set doesn't guarantee orientation, and its implications for gluing?

It seems to be that the formal definition of a Δ-set doesn't forbid identifying different faces of a simplex, that is, face maps $d_i$ and $d_j$, $i \ne j$ may map element $a : S_n$ to a same element $...
GolDDranks's user avatar
2 votes
1 answer
96 views

Affine map maps simplex onto a simplex

I am reading John Lee's Introduction to Topological Manifolds and I am trying to prove Theorem 5.39 (Simplicial Maps are determined by vertex maps. Let $K$ and $L$ be simplicial complexes. Suppose $...
nomadicmathematician's user avatar
0 votes
0 answers
57 views

Hatcher Simplicial homology [duplicate]

Im trying to solve a Problem from Hatcher: Compute the simplicial homology groups of the $\Delta$-complex obtained from n+1 simplices $\Delta_0^2,\Delta_1^2,...,\Delta_n^2$ by identifying all three ...
NoIdea's user avatar
  • 65
0 votes
0 answers
22 views

Connectivity of the flag complex of a ''union'' of two pentagons.

Suppose we have two graphs $\Gamma_1,\Gamma_2$ which are isomporhic to two pentagons and with vertices $v_1,\dots,v_5$ and $v_1',\dots,v_5'$ respectively. Let us construct a new graph $\Gamma$ with ...
Marcos's user avatar
  • 1,880
1 vote
1 answer
46 views

Number of Hole in a simplicial complex

I was going over Example 2.10 of Botnan, Magnus Bakke. "Topological Data Analysis Spring 2020.". I thought the number of holes is 3 which includes the hole from the linear combination of c1+...
Rowing0914's user avatar
1 vote
1 answer
79 views

Computation of the homology of a given semisimplicial set

I was given the following semisimplicial set. $X_0 = \{v_1, v_2\}, X_1 = \{a, b, c_1, c_2\}, X_2 = \{s, t\}, d_0(a) = d_0(b) = v_2, d_1(a) = d_1(b) = v_1, d_0(c_1) = d_1(c_1) = v_1, d_0(c_2) = d_1(c_2)...
idontknow's user avatar
0 votes
1 answer
92 views

Requesting help with Armstrong Topology Section 8.2 Problem 2

I'm working on Topology by Armstrong, and I'm struggling to complete problem 2 in section 8.2. The problem statement is as follows: Show the elementary 1-cycles, mentioned in Section 8.1, generate $...
Lukrau's user avatar
  • 75
2 votes
0 answers
60 views

References for Exercises on Abstract Simplicial Complexes

I am currently reading Combinatorial Algebraic Topology by Kozlov. I'm enjoying it a lot, however, it regretfully does not contain any exercises (it is more along the lines of lecture notes). In the ...
pyridoxal_trigeminus's user avatar
2 votes
0 answers
29 views

Is the Moore filler of a degenerate simplex itself?

In the proof that a simplicial group G is always Kan (see e.g. https://ncatlab.org/nlab/show/simplicial+group#AsKanComplexes ), you can always fill a $(m, j)$-horn $\lambda$ to a $m$-simplex $x$ with ...
Chenchang Zhu's user avatar
0 votes
1 answer
68 views

Do two triangulations of a smooth manifold have a common subdivision?

The Hauptvermutung (ie. the question in the title) is known to be false for PL manifolds and topological manifolds, but I can't find a result for smooth manifolds (with boundary), though I recall ...
JLA's user avatar
  • 6,484
3 votes
1 answer
44 views

$\mathbb{Q}$-acyclic simplicial complex which is not $\mathbb{Z}$-acyclic

I am reading preliminaries for a paper The worst way to collapse a simplex. It is written there: A [simplicial] complex $X$ is said to be $R$-acyclic [$R$ is a ring] if $H_i(X;R) = 0$ for all $i ≥ 0$. ...
brmbln's user avatar
  • 89
1 vote
0 answers
55 views

Why are there only finitely many simplicial maps from one polyhedron to another?

I don't understand why for two polyhedra $|X|$ and $|Y|$, there are finitely many choices of simplicial maps $$s: |X^m| \rightarrow |Y|$$ for some large enough $m \in \mathbb{N}$. Multiple sources say ...
Carson Newman's user avatar
2 votes
0 answers
61 views

Geometric Realization of an Abstract Simplicial Complex

I am currently working through this paper and something in the author's definition of geometric realisation of an abstract simplicial complex is unclear to me: (some stuff in between that is not ...
Sausage_Devourer's user avatar
0 votes
0 answers
15 views

Obstructions to existence of simplicial maps

Are there any simple obstructions for the existence of a simplicial map between two given simplicial complexes? A natural candidate would be some simple to compute function that is non-increasing ...
Kenneth Goodenough's user avatar
0 votes
1 answer
39 views

Subcomplex of the standard $n$-simplex having nonzero $H_{n-1}$

This is an additional exercise of Hatcher's Algebraic Topology book (link: https://pi.math.cornell.edu/~hatcher/AT/AT-exercises.pdf, section 2.1). Let $X$ be the standard $n$-simplex with its natural $...
user302934's user avatar
  • 1,570
0 votes
1 answer
25 views

Simplicial manifolds which do not satisfy Kan condition locally?

Similar to Kan condition for simplicial sets, there are also Kan condition for simplicial manifolds, that is, we ask the horn projection $p^k_j: X_k \to Hom(\Lambda[k,j], X)$ to be a surjective ...
Chenchang Zhu's user avatar
1 vote
2 answers
143 views

How to identify edges of $\Delta$-Complexes?

I have a question regarding exercise 2.1.6 in Hatcher: Compute the simplicial homology groups of the $\Delta$-complex obtained from $n+1$ $2$-simplices $\Delta_0^2,...,\Delta_n^2$ by identifying all ...
Henry T.'s user avatar
  • 1,324
0 votes
0 answers
30 views

Möbius function on a product of posets is the product of the Möbius function on each poset

I'm trying to prove that $\mu_{P\times Q}((p_1,q_1),(p_2,q_2))=\mu_{P}(p_1,p_2)\mu_{Q}(q_1,q_2)$, and my attempt is by induction. By definition, I get that: $\mu_{P\times Q}((p_1,q_1),(p_2,q_2))=-\...
Grimm Troupe's user avatar
0 votes
1 answer
50 views

infinite homotopy classes of maps between finite simplicial complexes

This post proves that there are at most countably many homotopy classes of maps $|K| → |L|$ for finite simplicial complexes $K$ and $L$. Can you give an example that there are countably infinite many ...
hbghlyj's user avatar
  • 2,780
2 votes
1 answer
59 views

Morphisms from simplicial sets into groupoids are determined by what?

For this question, one can assume that the simplicial sets come from oriented simplicial complexes (you can assume the orientation induces an orientation on the geometric realization). A morphism from ...
JLA's user avatar
  • 6,484
0 votes
0 answers
15 views

How the figures and vertices of simplicial complex are related to Hom([m],[n]) elements of standard n-simplex (trying to grasp Kan complex in it)?

Kan complex has nice pictorial interpretation in the simplicial complex, from the other side there is standard $n$-simplex. I am trying to understand how simplicial complex (and Kan horn in it and Kan ...
TomR's user avatar
  • 1,321
0 votes
1 answer
61 views

Definition of intersection of simplices

A singular $n$-simplex $\alpha: \Delta^n\rightarrow \Delta^p$ in $\Delta^p$ is called affine if $\alpha(\sum_{i=0}^n t_i e_i)= \sum_{i=0}^n t_i\alpha(e_i)$ holds for all $t_i$ with $\sum_{i=0}^n t_i=1$...
Margaret's user avatar
  • 1,777
3 votes
0 answers
35 views

Sufficient conditions for retractions along simplices to result in "nice" complex

Suppose we have a connected simplicial complex $X$ such that we can partition the vertex set of $X$ into disjoint $M_1,\ldots,M_k$ such that for all $i$, the induced simplicial complex on the vertices ...
marcelgoh's user avatar
  • 1,794
3 votes
0 answers
74 views

A Problem With Simplicial DeRham Cohomology

I am getting a contradiction by calculating the simplicial DeRham complex of an arbitrary manifold and getting it to be trivial. I also get a similar contradiction using the Godement resolution for ...
user127776's user avatar
  • 1,364
0 votes
1 answer
27 views

Removing simplices from a simplicial complex in sage

The context for this problem is this: given a simplicial complex $X$ and a simplex $\sigma \subseteq X$, I want to define a new simplicial complex $Y_\sigma$ to be the subcomplex of $X$ with vertex ...
Joe Wells's user avatar
  • 1,120
1 vote
1 answer
97 views

A matroid with zero Euler characteristic has an isthmus

Let $M$ be a rank $r$ matroid on the ground set $E$. $e\in E$ is said to be an isthmus of $M$ if $e$ is contained in every base of $M$. We also define the Euler characteristic of $M$ to be $$\chi(M)=\...
Diego Parodi's user avatar
28 votes
1 answer
2k views

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 ...
gwynneth-m.sc.'s user avatar
0 votes
0 answers
47 views

Understanding excision with another approach

I find a really interesting approach on Excision Axiom with subdivision operator and prism operator defined by a formula (without using induction). The material I followed is public and it has only 5 ...
Bogdan's user avatar
  • 1,899
3 votes
0 answers
62 views

Homotopy type of simplicial complex

I've encountered the following theorem (while reading some topology script): $\textbf{Theorem}$: Let $K$ be an abstract simplicial complex and $\{u, v\}$ it's edge. If link$_K \{u, v\}$ is ...
Brmbrm's user avatar
  • 71

1
2 3 4 5
12