Difference between revisions of "Algebraic topology"
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Note that the fundamental group is not in general [[abelian group|abelian]]. For example, the fundamental group of a figure eight is the [[free group]] on two [[generator]]s, which is not abelian. However, the fundamental group of a circle is <math>{\mathbb{Z}}</math>, which is abelian. | Note that the fundamental group is not in general [[abelian group|abelian]]. For example, the fundamental group of a figure eight is the [[free group]] on two [[generator]]s, which is not abelian. However, the fundamental group of a circle is <math>{\mathbb{Z}}</math>, which is abelian. | ||
| + | |||
| + | More generally, if <math>X</math> is an [[h-space]], then <math>\pi_1(X)</math> is abelian, | ||
| + | for there is a second multiplication on <math>\pi_1(X)</math> given by <math>(\alpha\beta)(t) = \alpha(t)\beta(t)</math>, which is "compatible" with the concatenation in the following respect: | ||
| + | |||
| + | We say that two binary operations <math>\circ, \cdot</math> on a | ||
| + | set <math>S</math> are compatible if, for every <math>a,b,c,d \in S</math>, | ||
| + | <cmath>(a \circ b) \cdot (c \circ d) = (a \cdot c) \circ (b \cdot d).</cmath> | ||
| + | |||
| + | If <math>\circ,\cdot</math> share the same unit <math>e</math> (such that <math>a \cdot e = e \cdot a = a \circ e = e \circ a = a</math>) then <math>\cdot = \circ</math> and both are abelian[http://abogadosenpalmademallorca.com .] | ||
== Higher Homotopy Groups == | == Higher Homotopy Groups == | ||
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== Homology and Cohomology == | == Homology and Cohomology == | ||
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{{stub}} | {{stub}} | ||
[[Category:Topology]] | [[Category:Topology]] | ||
Latest revision as of 14:35, 1 December 2015
Algebraic topology is the study of topology using methods from abstract algebra. In general, given a topological space, we can associate various algebraic objects, such as groups and rings.
Fundamental Groups
Perhaps the simplest object of study in algebraic topology is the fundamental group. Let 
be a path-connected topological space, and let 
be any point. Now consider all possible "loops" on that start and end at 
, i.e. all continuous functions ![$f:[0,1]\to X$](http://latex.artofproblemsolving.com/e/2/3/e23ac345d4603fbc14345da777eeece650ae1812.png)
with 
. Call this collection 
. Now define an equivalence relation 
on by saying that 
if there is a continuous function ![$g:[0,1]\times[0,1]\to X$](http://latex.artofproblemsolving.com/7/8/5/7856594758f5af46cda3cad2ccbf2bf8da35fdaf.png)
with 
, 
, and 
. We call 
a homotopy. Now define 
. That is, we equate any two elements of which are equivalent under .
Unsurprisingly, the fundamental group is a group. The identity is the equivalence class containing the map ![$1:[0,1]\to X$](http://latex.artofproblemsolving.com/7/d/a/7da75d7aadb2dddcfd6393787f20a4085a3c1e49.png)
given by 
for all ![$a\in[0,1]$](http://latex.artofproblemsolving.com/e/3/7/e3777765ea228b1655f3649991e57ddcc3253cdd.png)
. The inverse of a map 
is the map 
given by 
. We can compose maps as follows: 
One can check that this is indeed well-defined.
Note that the fundamental group is not in general abelian. For example, the fundamental group of a figure eight is the free group on two generators, which is not abelian. However, the fundamental group of a circle is 
, which is abelian.
More generally, if is an h-space, then 
is abelian,
for there is a second multiplication on given by 
, which is "compatible" with the concatenation in the following respect:
We say that two binary operations 
on a
set 
are compatible if, for every 
,
![\[(a \circ b) \cdot (c \circ d) = (a \cdot c) \circ (b \cdot d).\]](http://latex.artofproblemsolving.com/3/b/e/3bee0013dc1fe157bfbc4e968ec7a833e3513629.png)
If 
share the same unit 
(such that 
) then 
and both are abelian.
Higher Homotopy Groups
Homology and Cohomology
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