# Definition of singular n-simplex

Hatcher's book says 'A singular n-simplex in a space is by definition just a map $$\sigma:\Delta^n\rightarrow X$$'. And $$C_n(X)$$ be the free abelian group with basis the set of singular n-simplices in $$X$$.

So does he mean $$C_n(X)$$ is the... of all singular n-simplices in $$X$$?

And does he identify those 'same maps with different domain'? Because I see proposition $$2.8$$:

If $$X$$ is a single point, then $$H_n(X)=0$$ for $$n>0$$ and $$H_0(X)=Z$$.

The proof of this says:

In this case there is a unique singular n-simplex $$\sigma_n$$ for each n, and $$\partial(\sigma_n)=\sum(-1)^i\sigma_{n-1}$$, which is $$0$$ for n odd and $$\sigma_{n-1}$$ for n even...

But by definition of the boundary map: $$\partial_n(\sigma)= \sum(-1)^i \sigma|[v_0,v_1...\widehat{v_i},...v_n]$$. But different i deleted gives different domain, although they all map to a single point. This is what I mean 'same maps with different domain'.

Thanks for help!

• you precompose restriction with some map from delta^n-1
– bart
Mar 27 '20 at 9:12
• @bart Can you explain more? I don't understand your comment.. I'm new to the homology theory...
– ZWJ
Mar 27 '20 at 9:24
• sure, so in your boundary operator in the last paragraph, where you write the restriction map, thats actually the restriction precomposed with a map $\Delta^{n-1}\rightarrow \Delta^n$ given by mapping $v_j\rightarrow v_j$ for $j<i$ and $v_j\rightarrow v_{j+1}$ for $i<j$ (and extending linearly)
– bart
Mar 27 '20 at 9:42

$$C_n(X)$$ is the set of all formal sums of $$n-$$simplices in $$X$$, an element of $$C_n(X)$$ is given by a finite sum $$\sum n_i \phi_i$$ for some $$\phi_i : \Delta^n \rightarrow X$$ and $$n_i \in \mathbb Z$$. This is just a formal sum. And yes he means that $$C_n(X)$$ is the free abelian group generated by the set of $$\textbf{all}$$ n-simplices.
And that definition of the boundary map is really, really misleading, I could 100% see how you would think that deleting different $$i$$ give different domains. And according to that definition that is actually true. A better definition is $$\partial \sigma = \sum (-1)^i \sigma \circ d_i$$ where $$\sigma : \Delta^n \rightarrow X$$ and
$$d_i: \Delta^{n-1} \rightarrow \Delta^n : (x_0,...,x_{n-1}) \mapsto (x_0,...,x_{i-1}, 0,x_{i},...,x_{n-1})$$
then the resulting maps have the same domain and the resulting formal sum is in $$C_{n-1}(X)$$.
The map $$d_i$$ is the same as the map that @bart described in his comment by the way.