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Difference between revisions of "Algebraic topology"

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Perhaps the simplest object of study in algebraic topology is the [[fundamental group]]. Let <math>X</math> be a [[path-connected]] topological space, and let <math>x\in X</math> be any point. Now consider all possible "loops" on <math>X</math> that start and end at <math>x</math>, i.e. all [[continuous function]]s <math>f:[0,1]\to X</math> with <math>f(0)=f(1)=x</math>. Call this collection <math>L</math>. Now define an [[equivalence relation]] <math>\sim</math> on <math>L</math> by saying that <math>p\sim q</math> if there is a continuous function <math>g:[0,1]\times[0,1]\to X</math> with <math>g(a,0)=p(a)</math>, <math>g(a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now define <math>\pi_1(X)=L/\sim</math>. That is, we equate any two elements of <math>L</math> which are equivalent under <math>\sim</math>.
 
Perhaps the simplest object of study in algebraic topology is the [[fundamental group]]. Let <math>X</math> be a [[path-connected]] topological space, and let <math>x\in X</math> be any point. Now consider all possible "loops" on <math>X</math> that start and end at <math>x</math>, i.e. all [[continuous function]]s <math>f:[0,1]\to X</math> with <math>f(0)=f(1)=x</math>. Call this collection <math>L</math>. Now define an [[equivalence relation]] <math>\sim</math> on <math>L</math> by saying that <math>p\sim q</math> if there is a continuous function <math>g:[0,1]\times[0,1]\to X</math> with <math>g(a,0)=p(a)</math>, <math>g(a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now define <math>\pi_1(X)=L/\sim</math>. That is, we equate any two elements of <math>L</math> which are equivalent under <math>\sim</math>.
  
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Unsurprisingly, the fundamental group is a group. The [[identity]] is the [[equivalence class]] containing the map <math>1:[0,1]\to X</math> given by <math>1(a)=x</math> for all <math>a\in[0,1]</math>. The [[inverse]] of a map <math>h</math> is the map <math>h^{-1}</math> given by <math>h^{-1}(a)=h(1-a)</math>. We can compose maps as follows: <math>g\cdot h(a)=\begin{cases} g(2a) & 0\le a\le 1/2, \\ h(2a-1) & 1/2\le a\le 1.\end{cases}</math> One can check that this is indeed [[well-defined]].
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Unsurprisingly, the fundamental group is a group. The [[identity]] is the [[equivalence class]] containing the map <math>1:[0,1]\to X</math> given by <math>1(a)=x</math> for all <math>a\in[0,1]</math>. The [[Function/Introduction#The_Inverse_of_a_Function | inverse]] of a map <math>h</math> is the map <math>h^{-1}</math> given by <math>h^{-1}(a)=h(1-a)</math>. We can compose maps as follows: <math>g\cdot h(a)=\begin{cases} g(2a) & 0\le a\le 1/2, \\ h(2a-1) & 1/2\le a\le 1.\end{cases}</math> One can check that this is indeed [[well-defined]].
  
 
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.
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More generally, if <math>X</math> is an [[h-space]], then <math>\pi_1(X)</math> is abelian,
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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:
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We say that two binary operations <math>\circ, \cdot</math> on a
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set <math>S</math> are compatible if, for every <math>a,b,c,d \in S</math>,
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<cmath>(a \circ b) \cdot (c \circ d) = (a \cdot c) \circ (b \cdot d).</cmath>
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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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(I know next to nothing about these. Please fill in if you know about them.)
 
  
 
== Homology and Cohomology ==
 
== Homology and Cohomology ==
  
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(This is for when I'm feeling braver. Or, better yet, when someone else is feeling braver.)
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{{stub}}
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[[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 $X$ be a path-connected topological space, and let $x\in X$ be any point. Now consider all possible "loops" on $X$ that start and end at $x$, i.e. all continuous functions $f:[0,1]\to X$ with $f(0)=f(1)=x$. Call this collection $L$. Now define an equivalence relation $\sim$ on $L$ by saying that $p\sim q$ if there is a continuous function $g:[0,1]\times[0,1]\to X$ with $g(a,0)=p(a)$, $g(a,1)=q(a)$, and $g(0,b)=g(1,b)=x$. We call $g$ a homotopy. Now define $\pi_1(X)=L/\sim$. That is, we equate any two elements of $L$ which are equivalent under $\sim$.

Unsurprisingly, the fundamental group is a group. The identity is the equivalence class containing the map $1:[0,1]\to X$ given by $1(a)=x$ for all $a\in[0,1]$. The inverse of a map $h$ is the map $h^{-1}$ given by $h^{-1}(a)=h(1-a)$. We can compose maps as follows: $g\cdot h(a)=\begin{cases} g(2a) & 0\le a\le 1/2, \\ h(2a-1) & 1/2\le a\le 1.\end{cases}$ 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 ${\mathbb{Z}}$, which is abelian.

More generally, if $X$ is an h-space, then $\pi_1(X)$ is abelian, for there is a second multiplication on $\pi_1(X)$ given by $(\alpha\beta)(t) = \alpha(t)\beta(t)$, which is "compatible" with the concatenation in the following respect:

We say that two binary operations $\circ, \cdot$ on a set $S$ are compatible if, for every $a,b,c,d \in S$, \[(a \circ b) \cdot (c \circ d) = (a \cdot c) \circ (b \cdot d).\]

If $\circ,\cdot$ share the same unit $e$ (such that $a \cdot e = e \cdot a = a \circ e = e \circ a = a$) then $\cdot = \circ$ and both are abelian.

Higher Homotopy Groups

Homology and Cohomology

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