# From singular homology to 'regular' homology

Let $X$ be a topological space. The singular homology groups are the homology groups which arise from the singular chain complex. This is a chain complex

$\dots \longrightarrow C_k(X) \longrightarrow C_{k-1}(X) \longrightarrow \dots$

where each $C_k(X)$ is generated by the singular $k$-chains. A singular $k$-chain is a topological mapping from the standard $k$-simplex into $X$.

From this definition it is evident that a singular $k$-chain can be, well, singular. For example, it may map a high-dimensional simplex onto a single point.

What if we restrict the class of chains? I define $C_k^{\rm reg}(X)$ as the $\mathbb Z$-module generated by the topological embeddings of the standard simplex into $X$. When we consider the chain complex

$\dots \longrightarrow C_k^{\rm reg}(X) \longrightarrow C_{k-1}^{\rm reg}(X) \longrightarrow \dots$

then again we can compute the homology groups.

Question: How do these homology groups relate? Under which conditions is the embedding of the second complex into the first complex an isomorphism on homology?

This is a much more difficult chain complex to work with. If say $X$ is an $m$-manifold, then there will not be any "regular" simplices of dimension $>m$. Therefore in the "regular" homology, every $m$-cycle will give a distinct homology class. So say $H_1^{\text{reg}}(S^1)$ will be uncountable. We can divide up the usual generator of the homology of $S^1$ into a sum of two regular 1-chains in uncountably many ways and these will give distinct elements of $H_1^{\text{reg}}(S^1)$.