Define d(x,y) = d(H(x),H(y))
where H(x) denotes the sequence (H_q(x,eps)) for some eps fixed for now.
Say d(H(x),H(y)) = sum_q d(H_q(x),H_q(y)).
Recall H_q(x) = z + B_q(x) for all z in Z_q(x).
One way to define d(H_q(x),H_q(y)) is as |beta_q(x)-beta_q(y)| 
i.e. |rank(H_q(x))-rank(H_q(y))|.
The problem is that this ignores which cycles were involved in the cycles.
Idea: Just like we quotient out the cycles which are also boundaries, let's
also quotient out the cycles which are common. 
For example

        d-----b        versus        b-----f
         \   / \                    / \***/
          \ /   \                  /   \*/
           c-----a                c-----a

              x                     y

gives C_0(x) = <a,b,c,d>
      Z_0(x) = Ker (del_0 : C_0 -> C_{-1}) = C_0(x)
      B_0(x) = Im  (del_1 : C_1 -> C_0) = {0}
      H_0(x) = Z_0(x)/B_0(x) = C_0(x)/B_0(x) = <a,b,c,d>/{0} = <a+{0},b+{0},c+{0},d+{0}>

      C_0(y) = <a,b,c,f>
      Z_0(y) = Ker (del_0 : C_0 -> C_{-1}) = C_0(x)
      B_0(y) = Im  (del_1 : C_1 -> C_0) = {0}
      H_0(y) = Z_0(x)/B_0(x) = C_0(x)/B_0(x) = <a,b,c,f>/{0} = <a+{0},b+{0},c+{0},f+{0}>

      C_1(x) = <ab,ac,bc,bd,cd>
      Z_1(x) = Ker (del_1 : C_1 -> C_0) = <ab+bc+ca,bd+dc+cb>
      B_1(x) = Im  (del_2 : C_2 -> C_1) = Im (del_2 : {0} -> C_1) = {0}
      H_1(x) = Z_1(x)/B_1(x) = <ab+bc+ca,bd+dc+cb>/{0} 
                             = <ab+bc+ca+{0},bd+dc+cb+{0}>

      C_1(y) = <af,ac,fc,fd,cd>
      Z_1(y) = Ker (del_1 : C_1 -> C_0) = <af+fb+ba,ab+bc+ca>
      B_1(y) = Im  (del_2 : C_2 -> C_1) = Im (del_2 : <afb> -> C_1) = <fb-ab+af>
      H_1(y) = Z_1(y)/B_1(y) = <af+fb+ba,ab+bc+ca>/<fb-ab+af> 
                             = <af+fb+ba+<fb-ab+af>,ab+bc+ca+<fb-ab+af>>
                             = <0+<fb-ab+af>,ab+bc+ca+<fb-ab+af>>
                             = <ab+bc+ca+<fb-ab+af>>

Now we want to throw out the ab+bc+ca cycles because they are in common.
So we define
      
      Z'_1(x) = Z_1(x)/(Z_1(x) xsect Z_1(y))
              = <ab+bc+ca,bd+dc+cb>/(<ab+bc+ca,bd+dc+cb> xsect <af+fb+ba,ab+bc+ca>)
              = <ab+bc+ca,bd+dc+cb>/<ab+bc+ca>
              = <bd+dc+cb+<ab+bc+ca>>
      B'_1(x) = B_1(x)/(Z_1(x) xsect Z_1(y)) = {0}/<ab+bc+ca> = {0+<ab+bc+ca>}
      H'_1(x) = Z'_1(x)/B'_1(x) = <bd+dc+cb+<ab+bc+ca>>/{0+<ab+bc+ca>}
                                = <bd+dc+cb+0+<ab+bc+ca>>
                                = <bd+dc+cb+<ab+bc+ca>>

      Z'_1(y) = Z_1(y)/(Z_1(x) xsect Z_1(y))
              = <af+fb+ba,ab+bc+ca>/<ab+bc+ca>
              = <af+fb+ba+<ab+bc+ca>>
      B'_1(y) = B_1(y)/(Z_1(x) xsect Z_1(y)) 
              = <fb-ab+af>/<ab+bc+ca> 
              = <fb-ab+af+<ab+bc+ca>>
      H'_1(y) = Z'_1(y)/B'_1(y)
              = <af+fb+ba+<ab+bc+ca>>/<fb-ab+af+<ab+bc+ca>>
              = <(af+fb+ba+<ab+bc+ca>) + <fb-ab+af+<ab+bc+ca>> >
              = <0 + <fb-ab+af+<ab+bc+ca>> >
             ~= {0}