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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.

3 votes
0 answers
130 views

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. …
Marcel K. Goh's user avatar
1 vote

Amending flawed "proof" that homology groups are zero

. ▮ 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 …
Marcel K. Goh's user avatar
7 votes
4 answers
996 views

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 $\Delta_n$. …
Marcel K. Goh's user avatar