Difference between revisions of "07081300/Class notes for Tuesday, February 26"
(→A Homology Theory is a Monster) 

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'''Our Route:''' Axiom by axiom  state, apply, prove. Thus everything we will do will be, or should be, labeled either "'''S'''tate" or "'''P'''rove" or "'''A'''pply".  '''Our Route:''' Axiom by axiom  state, apply, prove. Thus everything we will do will be, or should be, labeled either "'''S'''tate" or "'''P'''rove" or "'''A'''pply".  
+  
+  ==Typed Notes==  
+  
+  <span style="color: red;">The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.</span>  
+  
+  ===First Hour===  
+  
+  
+  
+  Recall we had defined for a chain complex the associated homology groups: <math>H_p(C_*) :=\ ker\partial_p/im\partial_{p+1}</math>  
+  
+  From this we get the pth homology for a topological space <math>H_p(X)</math>  
+  
+  We have previously shown that  
+  
+  1) <math>H_p(\cup X) = \oplus H_p(X_i)</math> for disjoint unions of spaces <math>X_i</math>  
+  
+  2) <math>H_p(pt) = \mathbb{Z}\delta_{p0}</math>  
+  
+  3) <math>H_0(connected) = \mathbb{Z}</math>  
+  
+  4) <math>H_1(connected) \cong \pi_1^{ab}(X)</math> via the map  
+  
+  <math>\phi:[\gamma]_{\pi_1^{ab}}\mapsto[\gamma]_{H_1}</math>  
+  
+  <math>\psi:\sigma\in C_1\mapsto[\gamma_{\sigma(0)}\sigma\bar{\gamma_{\sigma(1)}}]</math> where <math>\sigma_x</math> is a path connecting <math>x_0</math> to x.  
+  
+  
+  We need to check the maps are in fact inverses of each other.  
+  
+  Lets consider <math>\psi\circ\phi</math>. We start with a closed path starting at <math>x_0</math> (thought of as in the fundamental group). <math>\phi</math> means we now think of it as a simplex in X with a point at <math>x_0</math>. <math>\Psi</math> now takes this to the path that parks at <math>x_0</math> for a third of the time, goes around the loop and then parks for the remaining third of the time. Clearly this is homotopic this composition is homotopic to the identity.  
+  
+  We now consider <math>\phi\circ\psi</math>. Start with just a path <math>\sigma</math>. Then <math>\psi</math> makes a loop adding two paths from the <math>x_0</math> to the start and finish of <math>\sigma</math> forming a triangular like closed loop. We think of this loop as <math>\sigma'\in C_1</math>  
+  
+  Now, we start from c being <math>c = \sum a_i\sigma_i</math> with <math>\partial c = 0</math>. So get <math>\sum a_i(\partial \sigma_1) = \sum a_i(\sigma_i(1)\sigma_i(0))</math>  
+  
+  So <math>\Psi(c) = [\gamma_{\sigma_i(0)}\sigma_i\bar{\gamma_{\sigma_i(1)}}]_{\pi_1}</math> which maps to, under <math>\phi</math>, <math>\sum a_i(\gamma_{\sigma_i(0)} + \sigma_i  \gamma_{\sigma_i(1)}) = \sum a_i\sigma_i = c</math> ( in the homology gamma's cancel as <math>\partial c = 0</math>)  
+  
+  
+  ''Axiomized Homology''  
+  
+  We now will move to an approach where we prove that our defined homology satisfies a series of established homology axioms that will allow us to apply the machinery of general homology to our specific "singular" homology defined via simplices.  
+  
+  Axiom 0) Homology if a function  
+  
+  '''Definition''' The "category of chain complexes" is a category whose objects are chain complexes (of abelian groups) and morphisms which is a homomorphism between each abelian group in one chain and the corresponding group in the other chain such that the resulting diagram commutes. I.e, <math>Mor((C_p)_{p=0}^{\infty}, (D_p)^{\infty}_{p=0}) = \{(f_p:C_p\rightarrow D_p)_{p=0}^{\infty}\ \ f_{p1}\partial_p^C=\partial_p^D f_p\}</math>  
+  
+  
+  Now, in our case, the chain complexes do in fact commute because <math>\partial</math> is defined by precomposition but f is defined by postcomposition. Hence, associativity of composition yields commutativity.  
+  
+  
+  '''Claim'''  
+  
+  Homology of chain complexes is a functor in the natural way. That is, if <math>f_p:C_p\rightarrow D_p</math> for each p induces the functor <math>f_*:H_p(C_*)\rightarrow H_p(D_*)</math>  
+  
+  The proof is by "diagram chasing". Well, let <math>c\in C_p</math>, <math>\partial c =0</math>.  
+  
+  Let <math>f*[c] = [fc]</math>. Now <math>\partial fc = f\partial c = 0</math>. Furthermore, suppose <math>c = \partial b</math>. Then, <math>f_*c = fc = \partial fb</math> so therefore <math>fc = \partial\beta</math> some <math>\beta</math>. This shows <math>f_*</math> is well defined.  
+  
+  Thus, for <math>c\in H_p(C_*)</math> get <math>fc\in H_p(D_*)</math> via the well defined functor <math>f_*</math>. 
Revision as of 21:59, 6 March 2008

A Homology Theory is a Monster
Bredon's Plan of Attack: State all, apply all, prove all.
Our Route: Axiom by axiom  state, apply, prove. Thus everything we will do will be, or should be, labeled either "State" or "Prove" or "Apply".
Typed Notes
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
First Hour
Recall we had defined for a chain complex the associated homology groups:
From this we get the pth homology for a topological space
We have previously shown that
1) for disjoint unions of spaces
2)
3)
4) via the map
where is a path connecting to x.
We need to check the maps are in fact inverses of each other.
Lets consider . We start with a closed path starting at (thought of as in the fundamental group). means we now think of it as a simplex in X with a point at . now takes this to the path that parks at for a third of the time, goes around the loop and then parks for the remaining third of the time. Clearly this is homotopic this composition is homotopic to the identity.
We now consider . Start with just a path . Then makes a loop adding two paths from the to the start and finish of forming a triangular like closed loop. We think of this loop as
Now, we start from c being with . So get
So which maps to, under , ( in the homology gamma's cancel as )
Axiomized Homology
We now will move to an approach where we prove that our defined homology satisfies a series of established homology axioms that will allow us to apply the machinery of general homology to our specific "singular" homology defined via simplices.
Axiom 0) Homology if a function
Definition The "category of chain complexes" is a category whose objects are chain complexes (of abelian groups) and morphisms which is a homomorphism between each abelian group in one chain and the corresponding group in the other chain such that the resulting diagram commutes. I.e,
Now, in our case, the chain complexes do in fact commute because is defined by precomposition but f is defined by postcomposition. Hence, associativity of composition yields commutativity.
Claim
Homology of chain complexes is a functor in the natural way. That is, if for each p induces the functor
The proof is by "diagram chasing". Well, let , .
Let . Now . Furthermore, suppose . Then, so therefore some . This shows is well defined.
Thus, for get via the well defined functor .