Difference between revisions of "Algebraic topology"
m (avoiding disambigs) |
m |
||
(4 intermediate revisions by 4 users not shown) | |||
Line 8: | Line 8: | ||
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 == | ||
− | |||
− | |||
== Homology and Cohomology == | == Homology and Cohomology == | ||
− | + | {{stub}} | |
+ | [[Category:Topology]] |
Latest revision as of 15:35, 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
with
. Call this collection
. Now define an equivalence relation
on
by saying that
if there is a continuous function
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 given by
for all
. 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
,
If share the same unit
(such that
) then
and both are abelian.
Higher Homotopy Groups
Homology and Cohomology
This article is a stub. Help us out by expanding it.