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I’m a little confused, for a chain to be acyclic, all Betti numbers must be zero. For a Betti number $\beta$

$\beta_i=\dim(Z_i)-\dim(B_i)$ where $Z_i=\ker(\partial_i)$ and $B_i$=im$(\partial_i{+}_1))$

I have $\beta_1=\dim(Z_1)-\dim(B_1)=2-3$, am I going about this the right way?

  • $\begingroup$ Isn't the image of the arrow $\mathbb{R}^2\to\mathbb{R}^3$ $2$-dimensional, not $3$-dimensional, if I'm interpreting your indexing correctly? $\endgroup$ – Ben West Dec 4 '18 at 1:03
  • $\begingroup$ Wouldn’t $B_1$=im$(T_2)=span{[1 -1 1]}$?, and that would be 3 dimensional? $\endgroup$ – LexyFidds Dec 4 '18 at 1:09
  • $\begingroup$ Aaaahhh you’re right, I think I’ve been indexing incorrectly $\endgroup$ – LexyFidds Dec 4 '18 at 1:12

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