Introduction to Complex Simplices: Why We Need Them

[Hymne]

Hello there!
Im trying to read Lee's Introduction to Topological Manifolds and got to the chapter of Simplices. What I missed in the introduction of this chapter is however a 'popular scientific' introduction of the use of these complec simplices - i.e. just some words about why we need them, which problems they solve etc.

Since the math is not to easy I feel I loose interest in these structures when I don't know a little bit in before.
So, could someone try to explain what the represent and why we need them?

Thanks!
Hymne

 

[quasar987]

So the previous chapters have introduced the notion of a topological space. The natural task that presents itself to the mathematician is then to classify the topological spaces. I.e., given two spaces, we want to be able to say if they are homeomorphic or not. Sometimes this is easy and an explicit homeomorphism can be constructed explicitely (ex: the open disk in R² is clearly homeomorphic to the R² plane itself via x-->x/(1-|x|)).

But other times, it won't be so clear whether 2 spaces are homeomorphic or not. In fact, topological spaces can be so complicated that it seems hopeless to find an algorithm that will be able to determine if two arbitrary spaces are homeomorphic. So what do we do when the problem at hand seems too hard? We first consider a simpler subcase. That is, we say "let us try instead to classify spaces that are 'nice' in some way or another, or that have some property in common."

One such class of space in which we can make progress on the classification problem is the class of "triangulable spaces" (or, put another way, the spaces that are homeomorphic to the geometric realization of a simplicial complex!) These are the spaces that can be constructed by gluing triangles (and their higher dimensional analogues, the so-called "simplices") along their edges (in some nice way). The triangulable spaces are nice spaces because they can be specified entirely (up to homeomorphism) by the "combinatorial data" consisting of
1) all the the triangles needed to build up the space
2) how they fit together.
(This combinatorial data is what is called a simplicial complex. It contains all the gluing information needed to construct the space called it geometric realization).

Unfortunately, there are (obviously!) many different ways to build a space up out of triangles in this way, so the combinatorial data corresponding to a given triangulation does not classify the triangulable spaces. That is, the map {simplicial complex}-->{its geometric realization}/homeomorphism is not injective.)... :(

However! It turns out that given a triangulation of a space, one can define certain properties of the combinatorial data that are (miracle!) independent of the particular choice of triangulation! This means in particular that these properties are "topological invariants". Namely: if P is such a property, and if X,Y are two triangulable spaces, then given any triangulations of these spaces, if X and Y are homeomorphic, then the triangulations both have property P or both don't have it. The useful corollary to this is that if you take two spaces and check for property P and one has it while the other does not, then you can conclude that they are not homeomorphic, and this is progress towards the classification problem.

So what are examples of such triangulation invariant properties? There is
1) Orientability: given a triangulation, can the triangles be given an "orientation" (clockwise or anticlockwise) in a "consistent way"? If so, the space is called orientable. So this is a simple invariant; it takes the value "YES" or "NO".

2) The Euler characteristic: call F the number of 2-dimensional triangles (faces), E the number of 1-dimensional triangles (edges) and V the number of 0-dimensional triangles (vertices) in a given triangulation T of X. Then the integer F-E+V is actually independent of T and so it is a characteristic of X itself called the Euler characteristic of X.

3) Simplicial homology: this is a powerful invariant because it is actually an infinite number of groups associated to a triangulation of the space, and different triangulations yield isomorphic groups. So given two triangulable spaces X and Y, if you compute the simplicial homology groups H_k(X) and H_k(Y) (k in Z) and you notice that for at least one index i in Z, the groups H_i(X) and H_i(Y) are not isomorphic, then X and Y are not homeomorphic.

Invariant 1) and 2) are introduced at the end of the chapter on simplicial complexes, while simplicial homology is explored in the very last chapter of the book. (At least in my edition of the book. Lee may have changed things around a bit in the 2nd edition.)

 

[quasar987]

Ah, and while it is true in general that the above invariants can only be used to tell spaces apart (two spaces may have the same orientability, the same euler char. and the same simplicial homology and still be non homeomorphic), it turns out that the data of orientability+euler char. completely classifies (victory!) the subclass of the class of triangulable spaces consisting of the so-called "closed surfaces". The details of this classification is the subject of the chapter called something like "the classification of surfaces".

 

[Hymne]

Thanks for a really good answer! You helped quite much. :)

One more question though. Considering the Euclidean simplicial complexes Lee starts out by choosing the number of k points in general position and then define our simplex by using the 1-norm.
In the next section the following sentence about the abstract simplicial complexes is found:

"Any element of \sigma \in K is called a vertex of \sigma..."

Is he here trying to say that the simplexes and vertexes are by no means unique for the abstract kind? I think I am confused here if this is not the case. A vertex could, by this definition, be an interior point of a Euclidean simplex - introduces before.

Concretley said: The manifold can be build up of many possible simlicies and vertex schemes... So any element in any simplex can be, and should be regarded as a vertex.

 

[quasar987]

Lee first defines Euclidean simplicial complexes as collections of actual triangles (simplices) lying in some large euclidean space R^N subject to some condition regarding their intersection.
So this is a very concrete concept and the link btw the Euclidean simplicial complex and the triangulable space it represents it clear: just take all the triangles in a Euclidean simplicial complex and consider their union and that's your space!

But notice that any n-simplex in R^N is entirely determined by its n+1 vertices since it is just the convex hull of these points. Hence a Euclidean simplicial complex is entirely determined by
1) the vertices of the simplices it contains
2) the data of which vertices belong to the same simplex
In other words, an equivalent way of specifying a Euclidean simplicial complex would be to specify a set whose elements are sets of points in R^N (the vertices). So this would look like { {a,b,c}, {a,b,d}, {d,e} , {a,b},{a,c}, {b,c}, {a,d} {a}, {b}, {c}, {d}, {e} }.

Then Lee goes on defining abstract simplicial complexes as precisely objects of this later type. That is, an abstract simplicial complex is a set whose elements are finite sets (to be thought of as representing some labeling of the vertices of a triangulable space).

So now an abstract simplex sigma is an element of an abstract simplicial complexe K and so it is just a fintie set {a,b,c,...,e}, where a, b, c,...e are of any nature whatesoever (not necessarily points in some euclidean space).

As an example of an abstract simplicial complex, consider a Euclidean simplical complex K and for every simplex sigma in K, form a set out of the vertices of sigma. The collection of all the sets thus obtained is an abstract simplicial complex.

Are you still confused?

 

[Hymne]

My confusion is certainly getting smaller and smaller.. ;)
I think I got it know. Thank you again!
Ill be back if there is some futher problem.:)

 

[Hymne]

I got it clearly!
Sumemrized it in swedish and I think I got a pretty good overall picture. The major problem I was dealing with was a stupid mistake, not paying attention to the word 'finite' in the definition of the abstract complex. I was thinking that the sets were not of the kind

{ {a,b,c}, {a,b,d}, {d,e} , {a,b},{a,c}, {b,c}, {a,d} {a}, {b}, {c}, {d}, {e} }

but rather

{ {All of the points in simplex #1}, {All of the points in simplex #2} ... }.

No wonder I got confused :)