## What is simplicial complexes in algebraic topology?

In algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices.

**Why are simplicial complexes important?**

Simplicial complexes were originally used to describe pre-existing topological spaces such as manifolds, as in the question. But now they are the key tool in constructing discrete models for topological spaces.

### How do you create a simplicial complex?

To build one, take the origin and 1 other point which lies on a coordinate axis. This construction, produces two 0-subsimplices. Next, connect the two points to get your 1-simplex σ1 = ⟨p0,p1⟩. 2-simplex (a simplex ⟨p0,p1,p2⟩ generated by three points, p0, p1, p2) A 2-simplex is a solid triangle (including its border).

**Is every simplicial complex a CW complex?**

Notably the geometric realization of every simplicial set, hence also of every groupoid, 2-groupoid, etc., is a CW complex. Milnor has argued that the category of spaces which are homotopy equivalent to CW-complexes, also called m-cofibrant spaces, is a convenient category of spaces for algebraic topology.

## What does CW complex stand for?

A CW complex is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory.

**What is a CW pair?**

In algebraic topology by a CW-pair (X,A) is meant a CW-complex X equipped with a sub-complex inclusion A↪X.

### Why are Betti numbers important?

In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.

**What is Betti curve?**

The Betti curve is a function mapping a persistence diagram to an integer-valued. curve, i.e. each Betti curve is a function B: R → N. Topological Data Analysis for Machine Learning. Bastian Rieck. 14th September 2020 14/30.

## Are simplicial complexes CW-complexes?

simplicial complex. Its n-skeleton Xn ⊂ X is formed by keeping only the i-simplices for i ≤ n. Since there is a homeomorphism (∆n,∂∆n) ∼= (DnSn−1), it is clear that X is a finite cw-complex, with one n-cell for each n-simplex. Despite appearances, simplicial complexes include many spaces of interest.

**Is every simplicial complex a CW-complex?**

### Is a CW complex a manifold?

Every compact smooth manifold admits a smooth triangulation and hence a CW-complex structure.