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11
votes
1answer
324 views

Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology?

There are numerous expositions of simplicial homology in the literature. Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes. Hatcher in “Algebraic ...
6
votes
1answer
194 views

Simplicial set are to cubical sets what simplicial complexes are to …?

Simplicial sets and cubical sets (with or without connections) are defined as presheaves over some indexing categories. There is a full subcategory of simplicial sets that we can identify with the ...
3
votes
1answer
155 views

Change of Betti numbers under simplicial maps

Let $\Delta$ be a simplicial complex on $n$ vertices, and $\phi$ a simplicial map that identifies two vertices $x$ and $y$ of $\Delta$. I want to show that the Betti numbers of $\phi(\Delta)$ cannot ...
2
votes
0answers
81 views

Inflation of $w_j(V_{SO(N)})$ and $w_j(M)$ from $SO(N)$ to $Spin(N)$ or Spin geometry

We know well this short exact sequence $$ 1 \to \mathbb{Z}_2 \to Spin(N) \to SO(N) \to 1. $$ The $j$-th Stiefel-Whitney class of the associated vector bundle of $SO(N)$, as $w_j(V_{SO(N)})$, can be ...
5
votes
0answers
140 views

Can we represent partitions by mutually parallel lines in the plane?

Lately I have become interested in the following idea: Suppose $n$ is a positive integer and $[n]=\{1,2,3,...,n\}$. Suppose we have 3 distinct partitions $b$, $g$, and $r$ of $[n]$. Assume that the ...
9
votes
1answer
280 views

Closed orientable surfaces have even Euler characteristic

It is of course completely standard that closed orientable surfaces have even Euler characteristic. What is the most elementary proof of this? More specifically, suppose I have a finite simplicial ...
2
votes
1answer
236 views

How does the high-dimensional combinatorial Laplacian work?

When considering the boundary and coboundary maps, we have the common definitions that the boundary map based on the space of chains $C_k(X)$ is $$\partial_k([v_0,...,v_k])=\sum_{i=0}^k (-1)^i[v_0,...,...
14
votes
0answers
185 views

How many cells are needed in a simplicial structure of $\mathbb{S}^n$ to induce all of $\pi_n(\mathbb{S}^m)$

Serre proved, that for (allmost) all $n,m\in\mathbb{N}$ the homotopy groups $\pi_n(\mathbb{S}^m)$ are finite, so - using simplicial approximation - for $n, m$ fixed there is a finite cell ...
2
votes
1answer
69 views

On the entries of a matrix representation for a boundary operator of a persistence module

In equation 6 of Computing Persistent Homology (page 8), the authors put forward the following identity: $$\deg \hat{e_i}+\deg M_k (i,j)=\deg e_j$$ Where $\hat{e_i}$ and $e_j$ are elements of ...
8
votes
1answer
233 views

Non-triangulable 4-manifold as a boundary of some 5 manifold

We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold. Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...
6
votes
2answers
439 views

Any 3-manifold can be realized as the boundary of a 4-manifold

We know "Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
3
votes
0answers
61 views

Have partition functions of abstract simplicial complexes been examined?

Many complicated probability distributions arising in electrical engineering and machine learning have a simple expression as a sum of products that can also be encoded in a factor graph. The ...
4
votes
1answer
124 views

Betti numbers of a Cohen-Macaulay Module in small projective dimension

I am trying to compute the Betti numbers of some Stanley-Reisner ring $R_\Delta$, where the underlying complex $\Delta$ is shellable and the projective dimension of the $R_\Delta$ is $3\text{ or }4$. ...
4
votes
1answer
68 views

Simplicial Pseudomanifolds with Boundary - Bounding number of maximal faces in terms of number of vertices and dimension

A simplicial pseudo-manifold of dimension $d$ with boundary is a simplicial complex satisfying the following conditions. Every maximal face has dimension $d$ Each face of dimension $d-1$ is a face ...
8
votes
3answers
386 views

Is there a discrete lattice analogue of conformal transformations?

There is a simple discrete combinatorial analogue of manifolds and homeomorphisms: Replace manifolds by simplicial complexes and homeomorphisms by Pachner moves. Equivalence classes of manifolds under ...
7
votes
2answers
205 views

Are there invariants of cell complexes similar to the Euler characteristic?

The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely \begin{equation} \chi=\...
12
votes
2answers
191 views

Vietoris-Rips complex Homology of a higher degree than the ambient dimension

Assuming we have a set of points $X=\{x_1,..,x_n\}$, all in $\mathbb{R}^d$, and construct the Vietoris-Rips-Complex $V_\epsilon (X)$ for some distance parameter $\epsilon > 0$. Is it possible to ...
9
votes
3answers
339 views

Minimal combinatorial data needed to define a polytope [duplicate]

Suppose I give a list of vertices $(v_1, v_2, ..., v_n)$, and a list of "adjacencies", i.e. pairs of vertices $(v_i,v_j)$. Does it exists a unique polytope that has this vertices and realises the ...
4
votes
0answers
92 views

Gaussian curvature/Euler characteristic of Facebook clusters

If I look at a connected subgraph on a small collection of actors (such as a small cluster) in the Facebook social network, and I find that 1) The Euler characteristic of the clique complex built on ...
0
votes
0answers
19 views

Describing a Simplicial complex of a regular polytope through the action of the automorphism group

This is from page 41 of "Abstract regular polytopes" Suppose that we have a group $\Gamma = \langle\sigma_1,\cdots,\sigma_n\rangle$ generated by involutions and denote $\Gamma_J = \langle\sigma_j\ \...
2
votes
0answers
58 views

Transfer map between simplicial manifolds

Let $M^m$ and $N^n$ be two triangulated oriented and closed manifolds and $f:M\to N$ a simplicial map. For each $a\in H_p(N)$ we may consider its homological transfer $f_!a\in H_{m-n+p}(M)$. I want ...
6
votes
0answers
122 views

Dowker and neighborhood complexes: reference wanted

Let $R$ be a 0-1 matrix whose rows or columns are maximal. Q1. Is there a name for such a matrix (or, e.g., a corresponding relation)? From 0-1 matrix corresponding to an abstract simplicial ...
1
vote
0answers
33 views

Criterion for bisemisimplicial sets to be a manifold

It is well known that a finite simplicial complex is a manifold of dimension $n$ iff the the link of each vertex is homeomorphic to $\mathbb{S}^{n-1}$. Are there any criteria for weaker structures? E....
4
votes
3answers
156 views

0-1 matrix corresponding to an abstract simplicial complex

Let $A$ be a 0-1 matrix whose columns are maximal. We can associate its rows with vertices and columns with simplices in an abstract simplicial complex. Conversely, given an ASC, we can encode it in a ...
1
vote
0answers
82 views

How to “intersect” or “refine” a pair of abstract simplicial complexes

Let $S,T$ be abstract simplicial complexes. Is there a (unique) abstract simplicial complex that gives me the most of what is in common with $S$ and $T$? I'm thinking of this as an "intersection," ...
1
vote
1answer
151 views

Approximating a compact $C^1$ hypersurface without boundary

Can we approximate (arbitrarily closely) a compact $C^1$ hypersurface in Euclidean space without boundary with a polygonal hypersurface, such as a simplicial complex? To clarify, I want to have the $\...
7
votes
2answers
189 views

Relation Degree of Dualizing Sheaf and Euler Characteristic

Let $C$ be a curve over $k$ and $w_C$ it's dualizing sheaf. If $dim_k H^0(C, \mathcal{O}_C) =1$ and $g:= H^0(C, \mathcal{O}_C)$ the arithmetic genus one easy computes $$deg(w_C) = 2g-2$$ where $deg$ ...
15
votes
3answers
843 views

Are finite spaces a model for finite CW-complexes?

Are finite topological spaces (i.e. topological spaces whose underlying set is finite) a model for the homotopy theory of finite simplicial sets (= homotopy theory of finite CW-complexes) ? Namely, ...
3
votes
1answer
244 views

Homotopy equivalence of geometric realizations

This question is related with this one. For simplicial complex (which we have to assume is ordered as explained in the answer of the linked question) we have a construction of geometric realization ...
4
votes
1answer
241 views

Homology of simplicial complex versus homology of simplicial _set_

Let $K$ be a simplicial complex: it consists from the set (called the set of vertices) and a family of subsets of set of vertices satisfying the property of being closed under taking subsets (those ...
5
votes
0answers
177 views

On a Robin Forman's remark on combinatorial simplicial complexes

In a very captivating introduction to discrete Morse theory, Robin Forman makes the following remark: ...However, that does not explain why so many simplicial complexes that arise in combinatorics ...
10
votes
1answer
204 views

Realisation of maps between spheres by simplicial maps

Let $K^n_0 := \partial \Delta_{n+1}$ the simplicial set obtained by removing the $(n+1)$-simplex from the standard simplex. This gives a simplicial decomposition of the sphere $S^n$. More generally, ...
4
votes
0answers
87 views

Calculating a certain parameter for an abstract simplicial complex

Let $\mathcal{C}$ be an abstract simplicial complex on some finite set $\Omega$. I say that a subset $\Lambda\subset\Omega$ is minimally non-simplicial if it is not a simplex, but all of its subsets ...
3
votes
0answers
98 views

Is the Evasiveness Conjecture strong enough to constructively imply the negation of the Pizzazz-conjecture?

Question. Assume the truth of the (notoriously open) Evasiveness Conjecture. Does this constructively imply the negation of the Pizzazz-conjecture? Remarks. The relevant statements are, by ...
1
vote
0answers
64 views

Every triangulation of an oriented matroid is partitionable

In his unpublished article, Combinatorial properties of triangulations of oriented matroids, Julian Webster proves that every triangulation of an oriented matroid is partitionable. Does this result ...
1
vote
1answer
179 views

Algorithm to check whether simplices intersect nicely

Suppose that $A$ and $B$ are both $3$-simplices linearly embedded in $\mathbb{R}^3$, say with vertices in $\mathbb{Q}^3$ so that we can do computations exactly. (I am also interested in the ...
7
votes
1answer
157 views

Fixed-point-free group action on a finite, contractible, 3-dimensional simplicial complex

Let $K$ be a finite simplicial complex with an admissible action of a finite group $G$. (Terminology: By an action of a group $G$ on $K$ I mean an action by simplicial automorphisms. The action is ...
2
votes
1answer
83 views

Polyhedral structure of functions writable as a finite signed sum of max of linear functions

For any two positive integers $k,n$ consider the space of functions writable as, $\sum_i \sigma_i \max \{ L_{i1},L_{i2},..,L_{ik} \}$ (a finite sum) where each $L_{*} : \mathbb{R}^n \rightarrow \...
7
votes
2answers
138 views

The number of $2$-simplices and the number of $1$-simplices in a $4$-dimensional simplicial complex

Given a $4d$ simplicial complex (a triangulation of $4$-manifold), is there any relation between the number of $2$-simplices (triangles) and the number of $1$-simplices (edges)? Generically, is the ...
10
votes
2answers
312 views

Is the configuration space of ordered triples of distinct points in the four-edge banana graph homotopy equivalent to a surface of genus 13?

If $X$ is a topological space, write $C_n(X)$ for the configuration space of distinct ordered tuples of points in $X$: $$C_n(X) = \{(x_1, \ldots, x_n) \in X^n \mbox{ so that $i \neq j \implies x_i \...
2
votes
1answer
93 views

Cyclic polytopes whose boundary is a flag complex

A cyclic polytope $C(n, d)$ is defined as the convex hull of $n$ distinct points on the moment curve in $\mathbb{R}^d$ (here $n>d$). This is a simplicial polytope so its boundary $\partial C(n, d)$ ...
1
vote
0answers
99 views

The lattice subgroup quotient

Let $G$ a finite group and $L(G)$ it's lattice of non trivial subgroups. Is it true that the quotient space $|L(G)|/G$ is contractible, where $|L(G)|$ is the geometric simplicial complex associated to ...
5
votes
0answers
113 views

torsion part of homology of simplicial complexes [duplicate]

Let $n$ be a fixed positive integer and let $K$ be a simplicial complex with $N$ vertices. Suppose the $n$-th integral homology group of $K$ is $$ H_n(K)=\mathbb{Z}^{\oplus i}\oplus (\oplus _{p \...
6
votes
1answer
298 views

Subcomplexes with homotopy type of a sphere in complexes with homotopy type of a wedge of spheres

Suppose $X$ is a finite $d$-dimensional simplicial complex which is homotopy equivalent to a wedge of at least two $d$-spheres. Does $X$ contain a subcomplex which is homotopy equivalent to a single $...
2
votes
2answers
161 views

Removing simplices from simplicial complexes without decreasing connectedness

Let $X$ be a non-contractible, $(d-1)$-connected, $d$-dimensional simplicial complex. By the theorems of Hurewicz and Whitehead, $X$ is homotopy equivalent to a wedge of $d$-spheres. Does there exist ...
2
votes
2answers
536 views

Intermediate between Vietoris-Rips complex and Cech Complex

The Vietoris-Rips complex (https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex) is an abstract simplicial complex that can be defined from any metric space M and distance $\delta$ by forming a ...
1
vote
0answers
56 views

Sampling in a polyhedral complex

Assume one is given a polyhedral complex $P$ in $\mathbb{R}^n$. Now consider picking uniformly at random a $D \subseteq \{0,1\}^n$. Is there way to upper bound the probability that $D$ (a subset of ...
7
votes
0answers
167 views

Shellable simplicial complex with restriction on shellings

Does there exist a (pure) shellable simplicial complex $\Delta$ with the following property? There is some facet $F$ of $\Delta$ such that no shelling can begin with $F$. This condition is easily ...
3
votes
1answer
174 views

Geometry of the second barycentric subdivision (and Thomason-fibrant replacement)

Is $\mathrm{sd}^2 (\Delta^n) = \mathrm{sd}^2(\partial \Delta^n) \times \Delta^1 \cup_{\mathrm{sd}^2(\partial \Delta^n) \times \{1\}} Cone(\mathrm{sd}^2(\partial \Delta^n))$ ? Here $\mathrm{sd}^2$ ...
8
votes
1answer
145 views

Algebraic Shifting Computer Code

Is anyone aware of computer code that will algebraically shift a simplicial complex (as in this Kalai paper)? Ideally, I am looking for an implementation that can run in something like Sage or ...