A question about algebraic topology
I was told that the method of algebraic topology losing many geometry information. What does this mean?
A good reference link will be more than enough.
- IvanLv 51 十年前最愛解答
The main difference between (classical) topology and algebraic topology, is the notion of " equal ".
Topology is the subject in which we study the open sets of the space.
Algebraic topology is the subject in which we assign algebraic objects to a space, e.g. groups, rings etc. The simplest topic is called fundamental group π_1. This group is something that classify the set of loops on the space, just like "classify how many method to wrap a rubber band on the space". So let me focus on this.
In topology, we say two space are *homeomorphic* if two space looks the same --- they have the same open set structure (same topology).
1) square <-> circle
2) cube <-> sphere
3) torus =/= sphere (torus has hole, sphere does not)
4) filled square =/= cube (one looks like a plane, one is 3D)
In algebraic topology, two space are " homotopic " if one can deform between spaces, as soon as the map is continuous, 可以壓得扁 is OK.
1) square <-> circle
2) filled square <-> cube (cube can be continuously 壓扁 to a filled square)
3) a plane with one point removed <-> a circle. (through the continuous map x-> x/|x|)
So definitely, we see that algebraic topology really lose geometry information --- a circle and a plane with a point removed are considered THE SAME!!!
so, why are we using " homotopic "? This is because
1) the algebraic object --- fundamental gorup --- of two homotopic space is exactly the same.
2) fundamental group is A LOT EASIER to calculate, then comparing open sets of different space as in usual topology.
3) if two space have different fundamental group, then they are not same (in the usual topology sense)
So we see that, algebraic topology, although losing some geometric information, will actually make it a lot easier to determine when two spaces are NOT the same (homeomorphic).
The converse is certainly not true: space with the same fundamental group may not be the same (as in the above examples).
The famous Poincare Conjecture is talking about this: In N dimension, if the fundamental group of a compact space is 0, then the space must be homeomorphic to the N-dim-sphere! So you can see that this statement is really hard to prove.
There are many more techniques in algebraic topology --- homology, cohomology, higher fundamental group, sheaves etc. Each one will lose some information, but will gain some unexpected result that other techniques are missing.
2007-07-23 18:19:27 補充：
Poincare Conjecture is positive.
you are right, sheaves is more on algebraic geometry. But ultimately they give topological invariants in homology theory, so anyway they are also algebraic topology.資料來源： PhD Math