Help with universal theorem of coefficients for homology

1. Sep 9, 2007

Coolphreak

*note before reading: pretend that the superscripts are really subscripts, for some reason , Latex is making the subscripts into superscripts*

I'm not sure if I'm interpreting the universal coefficient theorem for homology correctly. Let's say I have a Homology group H(X;Z). This is a homology group using ring of integers. Now, let's say I want to compute Homology over the Field $$Z$$$$_{2}$$ (integers modulo 2), in order to simplify matters. By the way, I am trying to calculate the Betti numbers for a simplicial complex. Now, let's say I have a matrix M whose nullspace is isomorphic to the homology group. Now, I want to make all of the entries in the matrix modulo 2, if possible, in order to greatly simplify calculations.

So far we have:
kerM = H(X; Z)

Now, the universal theorem gives us this: 0$$\rightarrow$$$$H$$$$_{k}$$($$X$$) $$\otimes$$G$$\rightarrow$$$$H$$$$_{k}$$(G)$$\rightarrow$$Tor(H$$_{k+1}$$($$Z$$),G)$$\rightarrow$$0

Basically, in this case, we can replace G with $$Z$$$$_{2}$$. Also, the torsion should go to zero, since we have a free group. Now, since kerM is isomorphic to $$H$$$$_{k}$$($$X$$), can I say that ker(M)$$\otimes$$ $$Z$$$$_{2}$$ is isomorphic to $$H$$($$X$$)$$\otimes$$ Z2= H(X;Z$$_{}2)$$$$Z$$$$_{2}$$ ? Can i also say that Ker(M) $$\otimes$$ Z$$_{}2$$ is the same as Ker(M$$\otimes$$Z$$_{}2$$) ?

Back to my goal again, I want to make the entries in the matrix M, integers modulo 2. However, I realize that the nullspace of M modulo 2 and and the nullspace of M are different. By applying the universal coefficient theorem, can I say that these two nullspaces are isomorphic? If this universal coefficient theorem is correct, can I just convert the entries of the new matrix ($$M$$$$\otimes$$$$Z$$$$_{2}$$) into 1's and 0's? It seems I would need to tensor the matrix $$M$$ with $$Z$$$$_{2}$$, but what would $$Z$$$$_{2}$$ in matrix form be? The identity matrix with modulo 2 entries? (basically just ones on the diagonal?). If I am totally wrong, can anyone suggest a method of making the nullspace of M and the nullspace of M modulo 2 isomorphic?

Last edited: Sep 10, 2007
2. Sep 10, 2007

Hurkyl

Staff Emeritus
You need to put consecutive symbols all within the same [ tex ] ... [ /tex ] tag to get the formatting right. To make it line up nicely with texts in paragraphs, you should use [ itex ] ... [ /itex ] instead of [ tex ] ... [ /tex ].