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Results tagged with homology
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user 168142
Homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
7
votes
4
answers
996
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Amending flawed "proof" that homology groups are zero
In the following, when we speak of homology of $\Delta_n$ and write $H_k(\Delta_n)$, we mean homology over ${\bf Z}$.
Proposition. For all $k\ge 1$, $H_k(\Delta_n)=0$.
"Proof". … The $k$th homology of this simplicial subcomplex is zero, so the $(k+2)$-tuple $(x_1,\ldots,x_{k+2})$ contributes nothing to the $k$th
homology of $\Delta_n$. …
- 411
1
vote
Amending flawed "proof" that homology groups are zero
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We'll also use the Mayer--Vietoris sequence, which for reduced homology states that
$$\eqalign{
\cdots \to H_s(A)\oplus H_s(B) \to H_s(X)\to H_{s-1}(A\cap B) \to
H_{s-1}(A)\oplus H_{s-1}(B)\to\cr …
- 807
3
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0
answers
130
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Homology groups of a certain simplicial complex
I've run across a simplicial complex which, according to Sage, seems to have a very easily-described homology. However, proving this fact has been rather difficult. … Then we need to prove that the other reduced homology groups are zero, which shouldn't be as difficult. …
- 807