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]

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### Trouble With the Definition of $\Delta$-Complex in Hatcher's Book

On Pg. 103 of Hatcher's Algebraic Topology the author has defined what a $\Delta$-complex is:
This is what I have gathered from what the author writes:
A $\Delta$-complex is a collection of oriented (geometric) simplices $\mathcal…

caffeinemachine

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### What is the statement of simplicial approximation theorem for homotopies?

In the wikipedia page simplicial approximation theorem and a former answer related to this theorem, it was mentioned that on simplicial complexes, homotopy between continuous mappings can be approximated using simplicial mappings and…

Wei Zhan

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### Classifying space infinite totally ordered set contractible

Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I expect it to be contractible.
However, I was not…

Tatjana Popow

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### (Co)homology question

I'm learning homology and cohomology by myself, and I've stumbled upon a nice introductory paper here (http://www3.nd.edu/~mbehren1/18.904/Heffern_project.pdf). On page 1, under Motivation, second paragraph, the authors say the following:
"We can…

boldbrandywine

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### cohomology of local system

Let $X_r$ be a finite simplical complex. Let $V_r$ be a sheaf which is a local system on $X_r$.
Is it true that:
$H^n(X_r,V_r$) i.e the cohomology of the sheaf $V_r$ coincide with the cohomology of the complex whose $n^{th}$ term is $\prod…

user100603

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### homotopy of simplicial maps between infinite complexes

I have such a problem.
Let $K,L$ be simplicial complexes, where $K$ is connected, countable and locally finite. In particular we can express $$K=\bigcup_{i=0}^\infty K_i,$$ where $K_0$ is some full and finite subcomplex (in particular compact) and…

pw1822

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