Representation of the fundamental class of a closed orientable $n-$manifold

Question

Let $M$ be a closed orientable $n$-manifold with a ∆-complex structure. Let ${σ_1 . . . , σ_k}$ be the set of all $n$-simplices. How does one prove that the fundamental class $[M]$ can be represented by (the cycle) $\sum\limits_{i=1}^{k}\sigma_i$ in simplicial homology?

Answer 1

Write down the simplicial complex. There are no $n+1$-cells, hence in dimension $n+1$ it is zero:

$$ \cdots \to 0 \to Z^\text{number of n-cells} \to Z^\text{something} \to \cdots$$

Now let there be $k$ n-cells. Then $ker = H_n(X) = Z^r \to Z^k$ gives you homology. It will be $Z^r$ for some $r = \#\pi_0(X)$. We can assume that $r=1$ and then you see that, since $M$ is orientable, the image of an $n$-simplex (ie its boundary) is precisely the negatively oriented boundary of its complement, ie $\partial \sigma_i = -\partial \sum_{i\neq j} \sigma _j$, hence, $\partial $ being a homomorphism $\partial \sum \sigma_i =0$.

An other way to see this (and this is by definition and this will be important for all fields where you need the fundamental class) is to take the characterising property of the fundamental class and check it. This will work in many different settings. It means check that the homology class restricts to local orientation. In the situation of a simplicial complex it is obvious to see on points in the interior of $n$-cells. Now think about how to fix it on the boundary...

Depending on your definition of orientation you can go with the first (which is basically the simplicial definition) or with the second (the local homology orientation definition).