Topology is an important branch of mathematics, which is related to the study of structures or the study of some surfaces of any object. Topology consists of the combination of two words, which came from Greek, one word is the “Topos” and the other word is “Logos”.
The meaning of Topos is the top or the upper surface of any shape and the meaning of Logos is the study, which is related to the study of surfaces of an object.

## Introduction to Algebraic Topology

Every surface of an object is always directly proportional to its geometrical object; hence, Topology is directly and closely related to Geometry too. In case of geometry, we have the shape of objects like squares, rectangle, line segments, and many more, with their specific properties like their areas, surface areas, or volume etc, but in case of Topology, we just study the surfaces of the above mentioned geometrical figures with their properties, which are known as Topological Properties.
Hence, Algebraic Topology is the study of topological properties of algebraic equations.
For example, consider the following diagrams:

Algebraic Topology helps in identifying the differences between the above shapes, from the algebraic curves as circles, hyperbolas or parabolas.
Algebraic Topology is mainly related to the study of algebraic topological properties like their continuous mapping in between their shapes, regardless of their distances.

We define an algebraic topology as a mapping which assigns an algebraic invariant to each and every topological space.
Mathematically, a topological space X is a continuous function in which any two mapping from S1: X could be continuously changed into one another. This is also known as a simply connected topology.
In algebraic topology we define a fundamental group $\pi_1$(X) as a directed loops inside that topological space. We define the loops as homotopic if and only if it has a common base point, continuously deforming each others loop.
example, consider the following diagram:

The above diagram shows an algebraic topology of the function f, g, h, i which are related to an initial point, as shown above. As the functions are related with associative property, therefore, the above algebraic topology is a homotopy.
Thus algebraic topology gives rise to a lot of designs and patterns, for the researchers for further exploring this branch of mathematics.

## Applied Algebraic Topology

Algebraic topology is widely applied nowadays, not in the field of mathematics but in the filed of science too, especially, physics, computer sciences and economics. Applied topology is also widely used for the topological and mathematical analysis of shapes, their sizes and very huge n-dimensional sets of data.

As we know that in algebraic topology we have to assign algebraic invariants to its topological given spaces, therefore, in applied algebraic topology, we apply this method to generate different functions resulting in various theorems, which are then further applied for various research purposes.
Consider, for example, the Euler characteristic, in which we assign all the integers to its topological spaces, which is a homotopy invariant. It is generally denoted by X.
For example, look at the following images:

 Name of Shape Image $\chi$ (Euler characteristic) Sphere 2 Torus 0 Mobius Strip 0

The next concept in applied algebraic topology includes the meaning of Simplicial Complex, which is a group of all the simplex (convex hull of all the points of a figure) kept together in such a way that the intersection of any two chosen simplicies results into a completely new simplex.

Regarding the above definitions and concepts, we define a Triangulation of the space as follows:
1).Each topological space maximum up to 3 dimensions is always homeomorphic to a simplicial complex.

2).Similarly, we can extend this definition in the field of vector spaces also, in which we can relate a vector space, on a field, mapped to n simplicies, where the base of the vector space is a simplex.
This concept of algebraic topology is applied to find the number of holes in a figure, which is the concept of simplicial homology.

3).Algebraic Topology can be applied by the method of Point Clouds, in which we collect points form a n dimensional space, and then prepare a topological method for solving it. In this method, we study a point cloud data by calculating its Hn(X) and then applying the Cech complex or Rips complex, whichever is applicable.

4).Another method for applied algebraic topology can be used as the Persistent Homology, in which a filtration F of a given simplicial complex X is calculated by making a sequence of its sub complexes in such a way that X0 is contained in X1 and so on up to Xn  which should be equal to X itself. We usually use the homology by making barcodes which shows its Betti numbers of its filtration.

## Differential forms in Algebraic Topology

Algebraic Topology has a wide connection with differential forms, which are very useful and important to study the various kinds of all differential forms, out of these the most important is, De Rham cohomolgy.
To study about the differential forms in algebraic topology we must know the following definitions and theorems:
1)
.Graded Algebras or Modules: A graded R module is a commutative ring R with identity element which is the left indexed collection of all the R- modules A = {An: n belongs to Integer.} Here, we can also use A(n) instead of An.

2).Concentrated in degree k: If A(n) = o for n not equal to k, then A is known as concentrated in degree k, denoted as A.

3).Non negative A: If A (n) = 0 for all negative number n, then A is known as non negative.

4).Sub-module: If A <= B, where A and B are R module, then A is the sub module of B.

5).Tensor Product: If A and B are R module, then their tensor product is:
(A x B)(n) = Tensor product from i + j = n over the ring R, of A(i) x B(j);

6).  Derivation Property: Let A is a graded algebra of degree one differential d, then the derivation property is defined as:
d(n) (xy) = d(p) (x) y + (-1)^p x d (q)(y)
7).Boundary  $\partial$: The boundary  $\partial$ of a k simplex is always equal to a chain of value (k - 1)-chain, and is calculated and denoted by the following formula:
$\partial$[p0, . . . , pk] = [Sigma from i = 1 to k] (-1)^i [p0, . . . , pi-1, pi+1, . . . , pk].
After computing we will find that  $\partial$ $\partial$ will always be equal to 0.
8).Calculation of Linearity of a chain: Now the boundary of a chain is always depends on its linearity; which is calculated as:
$\partial$
c = [Sigma i] ai * $\partial$c(from i to k)

Consider the following images, which form a De Rham complex:

Here, we can see that the boundary
$\partial$ of a k simplex is a chain of (k - 1). And the calculation of $\partial$ on a 3 cell shows that $\partial$ $\partial$c3 is equal to 0.

Similarly, we can now define the co boundary operator, which is denoted by $\delta$[p0, . . . , pk] = Sigma from p[p, p0, . . . , pk], where all the points [p, p0, . . . , pk] makes a simplex of value (k + 1), where in return it makes a co chain having all the (k + 1)
values, which consist of  the points [p0, . . . , pk] as the inclusion of their boundary. The calculation of $\delta$ on a one cell shows that that $\delta \delta$c1 is equal to 0.
This can be better understood by looking at the following diagram:

Hence, we found that there is an intense relation and connectivity between the general algebraic topology and differential forms.