 
 
 
 
 
 
 
What do they mean by "the Algebraic Topology conference is the nerd prom"??
 
That deserves a series of videos of Algebraic Topology
What is "Algebraic Topology"?
It is a branch of mathematics
which uses Algebra as a tool to solve
topological problems
But, what exactly is this tool?
If we consider the category of Topological Spaces
The tool which is named "functor"
associates to topological spaces certain
algebraic structures, which can be "groups",
or "rings", or "vector spaces"
or any other algebraic structure,
in such a way that to each topological space
it corresponds an algebraic object.
But recall that
in a category we also have maps
In particular, in the category of Topological Spaces
we have continuous maps, so we also have to associate
to each continuous map a homomorphism
between the algebraic objects
Moreover, this correspondence
must satisfies a couple of properties quite logical.
Rather easy to assume
First, if we have the identity map
between a topological space and itself
we must associate to this map the identity homomorphism
Second, if we have a second map
that can be composed with the first one
we associate to it a homomorphism
and the composition have to go
to the composition of the two homomorphisms
We are going to use these functors,
also called "topological invariants"
to solve problems in Topology.
The main problem in Topology is determine
if to spaces X and Y have the same shape,
that is,  if they are or not homeomorphic.
Recall that a homeomorphism between two topological spaces
is a bijective continuous map
whose inverse is also continuous,
that is, we have a couple of  maps f and g
such that if we compose them we obtain identities
The typical example of homeomorphism
is the one given between a donut and a cup of coffee
for this reason topologists always make the same joke
consisting in saying that a topologist does not distnguish
between a donut and cup of coffee at breakfast.
LOL
If we have a couple of homomorphisms between our topological spaces
we can apply our algebraic functor
to this diagram and we would obtain
a new diagram between algebraic objects.
This can be translated to the fact that
if X and Y are homeomorphic
then FX and FY are isomorphic
We usually use these functors
to give negative answers
to the question of two spaces
being homeomorphic, since, if for some reason
it is easier to work in the algebraic category
and we know that FX and FY
are NOT isomorphic, by the contrapositive argument
we will obtain that X and Y can not be homeomorphic
If we wish to give a positive answer
I mean,  if we want to prove that two spaces
ARE homeomorphic, we have no choice
but to build the homeomorphism
explicitely
Let's see an easy example! We are going to build
our first functor.
For that purpose, let me recall a definition
We say that a path in a topological space
is a continuous map from the unit interval
on our space
and we say that the path goes  from the point a
to the point b, if 0 is sent to a
and 1 is sent to b
We can define a relation
between the points of a topological space X.
We say that the point a is related
to the point b
if there is a path which goes from a to b
It turns out that this relation
is an equivalence relation. An we can prove it
checking the three properties
reflexive, symmetric and transitive.
The reflexive property states that
a is related to a
Indeed, we can define a path
which goes from the interval to the space
simply sending al the point of the interval to the point a
and we clearly see that this path
starts and end at a.
For the symmetric property we must check
that if a is related to b, then also
b is related to a
If we have a path which goes from a to b
we can reparametrize the path
and travel it in the opposite direction going from b to a
For the last one, the transitive property
which say that if a is related to b
and b is related to c, then a is related to c
Then, if we have a path from a to b
and another path going from b to c
we can define a new path from a to c
traveling at double speed the first path
and at double speed the second path.
Then, this relation  defines an equivalence relation
in the space X.
The equivalence classes for this relation
are called arc-connected components
of the space. For example,
for X the space of the picture, we have
three arc-connetcted components
We will also call Π sub zero of X
to the set of arc-connected components.
Moreover, if we have a second topological space Y
and a continuous map between them,
recall that a continuous map
send arc.connected components into arc-connected components.
Then, what we also have
is a map between sets
between the sets Π _0 (X)  and Π _0 (Y)
But looking carefully to what we have done, we have defined a functor
The functor   Π _0
which goes from the category of topological spaces to the category of sets
Of course, it is an algebraic category
with few structure, in fact with no structure at all, we have no operation
we simply have sets
And what we do is associate to a topological space X
the set of its arc-connected components
and as we have seen before, if we have a continuous map
f from X to Y
we associate a set map
So we can now apply
the criteria we have commented before
to determine that to spaces were not homeomophic
Here for example, we see that
The space X has
three arc-connected components
and the space Y has two arc-connected components
so the sets
Π _0 (X)  and Π _0 (Y) are not bijective
and therefore the spaces X
and Y can not be homeomorphic
With Π _0 we end our first video
of introduction to Algebraic Topology
and in the next video we will see that this functor
although it looks really simple, can be used
very interesting results. Bye!
 
 
 
