Intuition behind singular $n$-simplex

Question

I came across this definition that a singular $n$-simplex in a topological space $X$ is a continuous map $\sigma\colon \Delta^n \to X$. Using this definition a few examples were put forward: a singular $0$-simplex is a point, a singular $1$-simplex is an essential path in $X$. Now, I am aware that a $0$-simplex is a point and a $1$-simplex is a line. Can someone explain the similarities of these two examples?

Answer 1

The reason of considering singular simplices is because simplicial homology is only defined for simplicial complexes (or more generally $\Delta$-complexes). In that context the simplices are part of the underlying structure of the object you're working with, and using the properties of this structure it's clear how to define the "boundary" of a simplex as a formal sum of simplices of one dimension lower (being careful to take orientations into account so that the boundary operator is a differential on the chain groups).

But say we want to define a homology theory for ALL spaces. We like the combinatorial nature of simplicial homology and we want to replicate it as much as we can in the most general context, but there are no simplices given to us. So instead, where we would normally have a set of $k$-simplices given as part of the structure, the most natural thing to do is to say that "a simplex in $X$" is just a continuous function $\Delta^k \to X$. The "singular" part comes from the fact that these don't have to be embeddings, for example this includes constant functions. The "boundary" of a singular simplex is expressed as a formal sum of singular $(k-1)$-simplicies in an analogous way, by precomposing with the various simplicial face maps $\Delta^{k-1} \to \Delta^k$. Then to construct singular homology you consider free abelian group on the set of all continuous functions $$C_k^{sing}(X) = \mathbb{Z}\langle C(\Delta^k, X) \rangle = \mathbb{Z}\langle \{ \Delta^k \to X \} \rangle$$ with the appropriate boundary operator.

For a non-trivial example consider $\Delta^1\cong [0,1] \to S^1$ given by $t\mapsto e^{2\pi i t}$. This singular $1$-simplex is actually a $1$-cycle, and generates the singular homology group $H_1^{sing}(S^1)$. In fact this is part of a general example: for all $k>0$ the quotient space $\Delta^k/\partial \Delta^k$ is homeomorphic to $S^n$, and a "quotient map" $\Delta^k \to S^k$ will generate $H^{sing}_k(S^k)$.

Note: As continuous functions all singular simplices are automatically null-homotopic because $\Delta^k$ is contractible for all $k$, but this fine in terms of homology because the homotopy relation is not used at all in the definitions of our groups.