0708-1300/Class notes for Tuesday, February 26 - Revision history

Trefor at 18:34, 26 April 2008

2008-04-26T18:34:40Z

← Older revision		Revision as of 14:34, 26 April 2008
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	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 simplexes.		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 simplexes.

	Axiom 0) Homology if a ~~function~~		Axiom 0) Homology if a functor

	'''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_{p-1}\partial_p^C=\partial_p^D f_p\}</math>		'''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_{p-1}\partial_p^C=\partial_p^D f_p\}</math>
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	'''Definition'''		'''Definition'''

	Two morphisms <math>f,g:C_\rightarrow D_</math> between chain complexes are ''homotopic'' if you can find maps <math>h_p:C_p\rightarrow D_{p+1}</math> such that <math>f_p - g_p = \partial_{p+1} h_p = h_{p-1}\partial_p</math>		Two morphisms <math>f,g:C_\rightarrow D_</math> between chain complexes are ''homotopic'' if you can find maps <math>h_p:C_p\rightarrow D_{p+1}</math> such that <math>f_p - g_p = \partial_{p+1} h_p + h_{p-1}\partial_p</math>


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	Assume <math>[c]\in H_p(C_*)</math>, that is, <math>\partial c =0</math>		Assume <math>[c]\in H_p(C_*)</math>, that is, <math>\partial c =0</math>

	<math>[f_* c] - [g_* c] = [(f_-g_)] =[(\partial h + h\partial)c] = 0</math> (as <math>\partial c = 0</math> and homology ignores exact forms)		<math>[f_* c] - [g_* c] = [(f_-g_)c] =[(\partial h + h\partial)c] = 0</math> (as <math>\partial c = 0</math> and homology ignores exact forms)

	Hence, at the level of homology they are the same.		Hence, at the level of homology they are the same.
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	Define <math>h\sigma</math> = the above prism formed by <math>\sigma</math> and the homotopy H.		Define <math>h\sigma</math> = the above prism formed by <math>\sigma</math> and the homotopy H.

	<math>(f_-g_)\sigma = h\partial\sigma = \partial h\sigma</math>		<math>(f_-g_)\sigma = h\partial\sigma + \partial h\sigma</math>

	This, pictorially is correct but we need to be able to break up the prism, <math>\Delta_p\times I</math> into a union of images of simplexes.		This, pictorially is correct but we need to be able to break up the prism, <math>\Delta_p\times I</math> into a union of images of simplexes.

Trefor at 03:28, 8 March 2008

2008-03-08T03:28:24Z

@@ Line 22: / Line 22: @@
 We have previously shown that
-) <math>H_p(\cup X) = \oplus H_p(X_i)</math> for disjoint unions of spaces <math>X_i</math>
+''1) <math>H_p(\cup X) = \oplus H_p(X_i)</math> for disjoint unions of spaces <math>X_i</math>
 ) <math>H_p(pt) = \mathbb{Z}\delta_{p0}</math>
@@ Line 48: / Line 48: @@
 ''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.
+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 simplexes.
 Axiom 0) Homology if a function
@@ Line 67: / Line 67: @@
 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>.

Trefor: /* A Homology Theory is a Monster */

2008-03-07T02:59:44Z

A Homology Theory is a Monster

← Older revision		Revision as of 22:59, 6 March 2008
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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_{p-1}\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 pre-composition but f is defined by post-composition. 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>.

Drorbn at 00:07, 26 February 2008

2008-02-26T00:07:23Z

← Older revision		Revision as of 20:07, 25 February 2008
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	{{0708-1300/Navigation}}		{{0708-1300/Navigation}}
	{{In Preparation}}

	==A Homology Theory is a Monster==		==A Homology Theory is a Monster==

Drorbn at 00:05, 26 February 2008

2008-02-26T00:05:46Z

New page

{{0708-1300/Navigation}}
{{In Preparation}}

==A Homology Theory is a Monster==
[[Image:0708-1300-AxiomsForHomology.png|thumb|center|540px|Page 183 of Bredon's book]]

'''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 "'''S'''tate" or "'''P'''rove" or "'''A'''pply".