Algebraic Topology 101

Simply put, algebraic topology is about finding connections between abstract structures and topological spaces. Hence, we will start by introducing some abstract structures, in some details. We will then move on to introduce topology.  We will also understand where algebraic geometry, category theory, and homological algebra fit in.   Even though we assume that the reader is familiar with basic set theory, logic, Euclidean spaces and some other mathematical foundations, definitions will be introduced as needed.


  • Part 1- Abstract Structures

    • Group Theory

    • Linear Algebra

    • The structures in other branches of Maths

  • Part 2- Topological Spaces

  • Part 3- Homology-related Concepts

  • Part 4- Category Theory

  • Part 5- Applications

  • Resources


Part 1- Abstract Structures

Definition: An abstract structure consists of a set and one or more binary operations, satisfying certain axioms.

Example: An example of an abstract structure that everyone is familiar with is the set of real numbers, ℝ, with addition (+) and multiplication (×) as binary operations. The operations satisfy the following axioms:

1)    Associativity of (+) and (×)

2)   Commutativity of (+) and (×)

3)   Identity Element/elt

a) ∃! elt of ℝ called zero, denoted by 0, such that x + 0 = x, ∀ x. x ∈ℝ

b) ∃! elt of ℝ called one, denoted by 1, such that x × 1 = x, ∀ x. x ∈ ℝ

4)  Inverse

  a) ∀ x. x ∈ ℝ, ∃! y ∈ ℝ, such that x + y = 0

b) ∀ x ≠ 0. x ∈ ℝ, ∃! y ∈ ℝ, such that x × y = 1

5)   x × (y + z) = (x × y) + (x × z) for all x, y, z ∈ ℝ (Distributivity property)

In fact, every set with two binary operations satisfying the above axioms is an abstract structure called a field.

It would be an impossible endeavor to introduce every single abstract structure, as new ones are being introduced all the time. We will focus on groups and vector spaces.

Side Note:  An experienced reader might suggest that under certain conditions, a topological space could be considered an abstract structure. However, we will ignore this suggestion for now.

Part 1.1 Group Theory

Group theory is the study of groups. Group theory has many applications including ones in physics, chemistry and computer science. It took around a century for a group, as an abstract idea, to emerge. Independent bodies of work in, at least, permutations (Lagrange, Galois, Cauchy), number theory (Jordan, Kronecker) and Geometry (Klein, Lie) were linked and abstracted to give rise to the abstract structure, a group.     

  • Definitions of a Group, and an abelian group

  • Subgroups and Cosets

  • Generators, Cyclic and Free Groups

  • Homomorphism and Isomorphism

  • Applications

Groups, including Abelian Groups

Definition 1: A group is a non-empty set with one binary operation, let us denote it by ∗, satisfying the following axioms:

♣ If x ∈ G and y ∈ G then x∗ y ∈ G (closure)

♣ (x∗ y) ∗ z = x∗ (y ∗ z), for all x, y, z in G (associativity)

∃ element, e ∈ G, such that x ∗ e = e ∗ x = x, for all x ∈ G (identity)

♣ If x ∈ G, then ∃ element in G, called inverse and denoted by inv(x) such that x ∗ inv(x) = inv(x) ∗ x = e

The above implies that the identity e is unique in G, since if there is another e’ ∈ G with the same property, then e ∗ e’ = e’ ∗ e = e = e’. Also, the axioms imply that the inverse of each x ∈ G is unique. If an x in G has two inverses, inv1(x) and inv2(x), then inv1(x)= e ∗ inv1(x) = (inv2(x) ∗ x) ∗ inv1(x) = inv2(x) ∗ (x ∗ inv1(x)) = inv2(x) ∗ e = inv2(x).

Note that the group’s binary operation can be defined as a function:

G x G → G, denoted by (x, y) → x ∗ y and satisfying the above axioms - the closure property becomes redundant since being a function guarantees closure.

Examples: Some familiar examples to think of:

1)   The set of integers, ℤ, with the addition operator

2)   ℤ is not a group under multiplication

3)   The set of real numbers, ℝ, under the addition operator

4)   The set G = {0, 1, 2, …., n-1} of integers modulo n is a group.

5)   The set of 2x2 matrices with real entries such that the determinant is non-zero, is a group under usual multiplication or composition of matrices. This group is denoted by GL(2, ℝ) and called general linear group of degree 2. In general, GL(n, ℝ) is a group consisting of invertible (i.e. determinant is non-zero) matrices with real entries.

A variety of interesting groups will be discovered as we delve into more Mathematics. For example, once we are familiar with manifolds, we can introduce a Lie group.

Definition 2: An abelian group is a group whose binary operation is commutative.

Examples of abelian groups:

▶ The set of integers, ℤ, with the addition operator

▶ The set of real numbers excluding 0, ℝ*, with the multiplication operator

▶ The set 2^ defined as {2^r: r ∈ } = {1, 2, 1/2 , 4, 1/4 , 8, 1/8 , . . . }, where p^q is p to the power q, is an abelian group under the multiplication operator.

Since a group is a set then one can talk about subsets.

Subgroups and Cosets

Definition 3: A subset H of a group G is a subgroup of G iff H is a group under the binary operation of G.

It follows that:

♦ The singleton identity element, {e}, is a subgroup of G under any operation.

♦ G is a subgroup of itself

These two subgroups are called improper subgroups. Finding proper subgroups is an important issue in group theory.

Definition 4: If H is a subgroup of a group G, and g ∈ G, then the right coset of H formed by g is

H ∗ g = { h ∗ g, where h ∈ H } and,

the left coset of H generated by g is

g ∗ H = { g ∗ h, where h ∈ H }.

It follows that in an abelian group, there is no need to distinguish between a right coset and a left coset. A coset does not have to be a subgroup of G. However, if g ∈ H then the cosets are subgroups of G.  

Definition 5: The order of a group G, denoted by |G|, is the cardinality of G.  

Example: The singleton group G={e, where e is the identity element} is the only group of order 1 such as G={0} under addition or G = {1} under multiplication. In this case, G is called a trivial group.

Side Note: There is a special type of groups considered the building blocks for all groups, similarly to elements in chemistry or prime numbers in number theory. These building blocks are called simple groups. A simple group is defined as a nontrivial group whose only normal subgroups are the trivial group and the group itself. Mathematicians worked on the discovery and classification of finite simple groups for more than 30 years. The proof they provided for that topic affected many areas in Mathematics. We will define a normal subgroup and re-visit the simple groups at a later time.  

Generators, Cyclic and Free Groups

Before we introduce generators of a group, we need to introduce general exponentiation.

Definition 6: Let x ∈ G, y its inverse and t ∈ then

xt =

  • x if t = 1, y if t = -1
  • e if t = 0 and
  • x ∗ xt-1 if t > 1, y ∗ y|t|-1 if t < -1

Definition 7: S ⊂ of a group G is a generating set of G iff ∀ x. x ∈ G,

x= s1a1 ∗ s2a2 ∗ s3a3 ∗ … ∗  srar  for some r ∈ ℤ, where si ∈ S and ai ∈ ℤ  for i=1, 2,…, r. The elements of S are called generators. To denote that S generates G, we write G = < S >.

Definition 8: a cyclic group is a group generated by a singleton.

Definition 9: The order of an element, g, of a group G, is the least positive integer n, such that, g^n=e. If no such integer exists, the order of g is said to be infinite. We denote the order of g by |g|.

Definition 10: a free group is a group where there is no relation between its generators except the one required by a group’s axiom about inverses.

A reader would have some related questions after the last few definitions:

♦ A generation is a relation but a relation does not have to be a generation, is that true?

♦ Can elements of S generate each other i.e. written in terms of each other as exponentiation?

♦ If elements of S can generate each other, does that mean there is more than one set and can one talk about a smallest generating set?

♦ Does every group have a generating set?

♦ If a group has a generating set, is it unique?

♦ Is the generating set finite?

♦ Is it possible that the generating set itself is a subgroup of G?

♦ Do inverses of elements of S belong to S?

Before we answer the above questions, we will introduce some more concepts.

Isomorphism and Homomorphism

Definition 11: Let G1 be a group with operation ∗, and G2 be a group with operation ο. If there is a bijection, Φ, between G1 and G2 such that for all x, y ∈ G1 the following holds:

Φ (x ∗ y) = Φ (x) ο Φ (y)

then (G1, ∗) and (G2, ο) are called isomorphic, denoted by G1 ≅ G2, G1 ≈ G2 or even G1 ∼ G2, and the mapping Φ is called an isomorphism.

An isomorphism Φ: G → G is called an automorphism, that is, an isomorphism of a group to itself.  

More generally,

If there is a function, Ω, between (G1, ∗) and (G2, ο) satisfying the above condition then Ω is a homomorphism and (G1, ∗) and (G2, ο) are said to be homomorphic to each other.

To make it clearer, in order to show that two groups (G, ∗) and (H, ο) are isomorphic, we need to ensure that there is a one-to-one correspondence and that the operation on each group is preserved:

1)     Define a function (mapping) Φ: G → H

2)    Prove that Φ is injective i.e. Φ(x) = Φ(y) ⇒ x=y, for all x, y in G

3)    Prove that Φ is surjective i.e. ∀ h. h ∈ H, h = Φ(g), for some g ∈ G

4)    Show that Φ preserves the group operations i.e.

Φ (x ∗ y) = Φ(x) ο Φ(y)

What does all that mean in “plain” terms?

Let us start with examples.

Examples of Isomorphic Groups:

1)     Consider G = (ℝ, +) and H = (ℝ*, x). G ≅ H.

a.     Define Φ: G → H as Φ(g)= 2^g where g ∈ ℝ

b.     One can easily show that Φ is a bijection

c.      The operation-preserving equality holds:

Φ(g1+g2) = 2^(g1+g2) = (2^g1) x (2^g2) = Φ(g1) x Φ(g2)

       In fact, any Φ: G → H as Φ(g)= a^g where a ∈ ℝ* and g ∈ ℝ, would be a good choice.

2)    Consider G = (, x) and H = (2^ℤ, +). G ≅ H.

The same as above with Φ: G → H as Φ(g)= 2^g where g ∈

3)    Consider G = SL(2, ℝ) with usual matrix multiplication. G is automorphic.

a.     Define Φ: SL(2, ℝ)  → SL(2, ℝ) as Φ(A) = M A M-1 where M is a constant matrix that was picked from G and A is any element, variable, in G.

b.     Φ is injective:  Φ(A) = Φ(B) ⇒ M A M-1 = M B M-1   ⇒   M-1 M A M-1 M = M-1 M B M-1 M ⇒ A = B.

c.      Φ is surjective: Let A ∈ SL(2, ℝ), I A I = A where I is the identity element of SL(2, ℝ). This implies that (M M-1 ) A (M M-1) =  M (M-1  A M) M-1 = M B M-1 = A, where B= M-1  A M. Since A ∈ SL(2, ℝ) then M-1  A M ∈ SL(2, ℝ).

d. Operation Preservation holds: Φ(AB) = M AB M-1 and Φ(A)Φ(B) = (M A M-1) (M B M-1) = M A (M-1 M) B M-1 = M A B M-1. Hence, Φ(AB) = Φ(A)Φ(B).

As one can see from the examples, isomorphisms preserve the structure of the elements of two groups. The image under Φ: G → H of the product of two elements is the same as the product of the images of each element. The image of the neutral element of the domain of Φ, G, is the same as the neutral element of the range of Φ, H.  Thus, the only difference between G and H are that the elements as well as the operations have different names; otherwise, they are the same structure.

Hence, isomorphism is really a very powerful tool. It determines whether two groups have similar properties. Why is this important?

Because if one knows that two groups G and H have similar properties and s/he knows all the properties of G then one would know all the properties of H.

What about homomorphisms? Are they as powerful as isomorphisms?

Not as powerful but they are also useful in classifying algebraic structures since if two algebraic systems/structures are homomorphic then some features of one structure can be applied to the other without having to analyze these features on the second structure. Homomorphism identifies how related the different algebraic structures, in this case, groups, are.


Applications

Groups have applications in chemistry, particle physics, computer science and other branches of Mathematics such as algebraic geometry and number theory. In the following, we are going to focus only on applications in cryptography and robotics. The following is only a tip of the iceberg.  

  • Group Theory in Crypto-systems

  • Group Theory in Robotics


Group Theory in Crypto-systems

For an introduction on cryptology and crypto-systems please see corresponding wiki-page. Here, we assume the reader is familiar with the basics.

Though modern cryptosystems deal with multi-party communication, we restrict our scenario to the well-known two-party Alice and Bob, not necessarily humans, wanting to communicate securely over an insecure channel.

To do so, Alice and Bob would have to have a shared information, a secret shared code or key that they only know. If they have that and they know how to cipher it then they should be able to communicate securely as long as the “enemy” or eavesdropper cannot figure out the key.

One protocol they could use to generate a shared secret key and exchange that key is called Diffie-Hellman, referring to the two researchers who proposed it. The Diffie-Hellman algorithm is based on algebraic structures and, as we will see, it depends on finding a group or a field that makes solving a certain equation intractable.

[[Note on what is considered group-based cryptosystems in the literature]]







 

Resources

The following list of resources, for this wikipage, is in no particular order.