On pp.108-109 of the 2015 version of his Algebraic Topology, when introducing singular homology Professor Hatcher says some (to me) intriguing things about the relationship between singular simplices and manifolds, which unfortunately are neither proven nor given a reference.
Does anyone know of references verifying or discussing in further detail the statements made?
I've already looked at Fenn's Techniques of Geometric Topology, and Kreck's Differential Algebraic Topology, which both seem to describe similar phenomena, but I don't understand if either of the two texts is talking about the same thing as Hatcher or if I am confusing what they discuss for something else. (Kreck's book is also freely available on the internet.) Specifically, Fenn talks about "geometric chains" and "manifolds with singularities codimension $\le n-2$", while Kreck talks about stratified generalizations of $n$-manifolds called "$n$-stratifolds" which have empty $(n-1)$-stratum (in general I think), but not necessarily empty strata of dimension $(n-2)$ or lower.
It all sort of seems related, but Hatcher doesn't give any references, and none of the three books uses the same terminology, so I am not actually sure if they are all referring to the same concepts or not. In particular, Kreck defines a new homology theory using stratifolds which he shows to be different from singular homology for some spaces not homotopy equivalent to CW complexes, while Hatcher and Fenn both seem to be discussing re-interpretations of singular homology.
(The quote/excerpt from Hatcher is below.)
Cycles in singular homology are defined algebraically, but they can be given a somewhat more geometric interpretation in terms of maps from finite $\Delta$-complexes. To see this, note first that a singular $n$-chain $\xi$ can always be written in the form $\sum_i \varepsilon_i \sigma_i$ with $\varepsilon_i = \pm 1$, allowing repetitions of the singular $n$-simplices $\sigma_i$. Given such an $n$-chain $\xi = \sum_i \varepsilon_i \sigma_i$, when we compute $\partial \xi$ as a sum of singular $(n-1)$-simplices with signs $\pm 1$, there may be some canceling pairs consisting of two identical singular $(n-1)$-simplices with opposite signs. Choosing a maximal collection of such canceling pairs, construct an $n$-dimensional $\Delta$-complex $K_{\xi}$ from a disjoint union of $n$-simplices $\Delta_i^n$, one for each $\sigma_i$, by identifying the pairs of $(n-1)$-dimensional faces corresponding to the chosen canceling pairs. The $\sigma_i$'s then induce a map $K_{\xi} \to X$. If $\xi$ is a cycle, all the $(n-1)$-dimensional faces of the $\Delta_i^n$'s are identified in pairs. Thus $K_{\xi}$ is a manifold, locally homeomorphic to $\mathbb{R}^n$, near all points in the complement of the $(n-2)$-skeleton $K_{\xi}^{n-2}$ of $K_{\xi}$. All the $n$-simplices of $K_{\xi}$ can be coherently oriented by taking the signs of the $\sigma_i$'s into account, so $K_{\xi} \setminus K_{\xi}^{n-2}$ is actually an oriented manifold. A closer inspection shows that $K_{\xi}$ is also a manifold near points in the interiors of $(n-2)$-simplices, so the nonmanifold points of $K_{\xi}$ in fact lie in the $(n-3)$-skeleton. However, near the points in the interiors of $(n-3)$-simplices it can very well happen that $K_{\xi}$ is not a manifold.
In particular, elements of $H_1(X)$ are represented by collections of oriented loops in $X$, and elements of $H_2(X)$ are represented by maps of closed oriented surfaces into $X$. With a bit more work it can be shown that an oriented $1$-cycle $\bigsqcup_{\alpha} S_{\alpha}^1 \to X$ is zero in $H_1(X)$ if and only if it extends to a map of a compact oriented surface with boundary $\bigsqcup_{\alpha} S_{\alpha}^1$ into $X$. The analogous statement for $2$-cycles is also true. In the early days of homology theory it may have been believed, or at least hoped, that this close connection with manifolds continued in all higher dimensions, but this has turned out not to be the case. There is a sort of homology theory built from manifolds, called bordism, but it is quite a bit more complicated that the homology theory we are studying here.
Is the failure of these properties perhaps related to the non-existence of triangulations for some $4$-dimensional manifolds, for example? This passage raises a lot more questions for me than it answers, so I really wish there were more references to further investigate the questions raised.
Note: to be fair to Fenn, on pp. 9-12, he seems to discuss something similar to that discussed by Hatcher, although I am not sure if it is truly the same concept or not -- he mentions something called a "geometric chain" which Hatcher does not. The relevant excerpt (from Fenn) is below:
The original idea of homology due to Poincare, Veblen, and others was to make an algebra based on manifolds. It would indeed be pleasant if all homology classes could be represented by manifolds. However, theory has shown this to be false, although it is possible to represent homology classes by something which is almost a manifold.
Definition 1.3.5. Geometric chains.
An $n$-circuit (or manifold with codimension $2$ singularities) is a complex $K$ with the following properties:
Every point of $K$ either lies in an $(n-1)$-cell or in the interior of an $n$-cell.
Every $n$-cell and $(n-1)$-cell is homeomorphic to a ball.
An $(n-1)$-cell is the face of one or two $n$-cells. The closure of the $(n-1)$-cells satisfying the former condition is a subcomplex of $\partial K$, the boundary of $K$.
The $n$-cells of $K$ are oriented to form an integer chain $c$ such that $|\partial c| = |partial K|$.
The boundary $\partial K$ is an $(n-1)$-circuit.
So the complex $K$ is an $n$-manifold except for a sub-complex (the singularities) of dimension $\le n-2$.
A geometric $n$-chain in a space $X$ consists of an $n$-circuit $K$ and a map $f: K \to X$.
If $f(\partial K) \subset A$ then $f:(K, partial K) \to (X,A)$ represents some class in $H_n(X,A)$. Before considering the converse theorem it is necessary to recall the definitions of singular homology...
The singular theory is totally useless as a calculating tool, but it is very convenient theoretical device, as the following theorem shows:
Theorem 1.3.7. Any singular homology class in $H_n(X,A)$ can be represented by a geometric $n$-chain $f: (K, \partial K) \to (X,A)$.
The idea behind the proof is to use a singular chain representing the class to define the geometric chain.
Let the class be represented by the singular chain $$c = \sum_{i=1}^r \lambda_i f_i \,\,, $$ where the $\lambda_i$ are non-zero integers and the $f_i$ are singular $n$-simplexes in $X$.
Let $Y$ be the disjoint union of $|\lambda_1| + |\lambda_2| + \dots + |\lambda_r|$ $n$-simplexes. Partition $Y$ into $k$ blocks $\{ B_i \}_{i=1}^k$ where $B_i$ contains $|\lambda_i|$ $n$-simplexes, $i=1,\dots, r$. For each $n$-simplex $\sigma$ in $Y$ let $a_{\sigma}: \sigma \to \Delta_n$ be a simplicial isomorphism matching $\sigma$ with the standard $n$-simplex. Consider the collection of all pairs $(\sigma, s)$, where $\sigma$ is an $n$-simplex of $Y$ and $s$ is a face of $\sigma$. Call two pairs $(\sigma, s)$ and $(\tau, t)$ identifiable if the following conditions hold:
- If $\sigma \in B_i$, $\tau \in B_j$ then $i \not= j$.
- If $a_{\sigma}(s)$ is the face of $\Delta_n$ not containing $v_k$ and $a_{\tau}(t)$ is the face not containing $v_{\ell}$, then $f_{i(k)} = f_{j(\ell)}$.
- $(-1)^{k + \ell} \lambda_i \lambda_j < 0$.
The last condition ensures that the contribution of the faces $s$ and $t$ in $c$ cancel out when the boundary is taken.
From $Y$ a manifold with singularities is built up by identifying $(n-1)$-simplexes. Pick two identifiable pairs $(\sigma,s)$ and $(\tau,t)$ in $Y$ and using the notation above identify $s$ and $t$ by the rule $ x \sim y$ if $\partial_k^{-1} a_{\sigma}(x) = \partial_{\ell}^{-1} a_{\tau}(y)$. This reduces the number of identifiable pairs by one. Continue in this fashion until none is left. The resulting complex $K$ is a circuit. Let $f: K \to X$ be defined on $n$-cells by the compositions $f_i a_{\sigma}$. It is not hard to see that $f: (K, \partial K) \to (X,A)$ corresponds to the conditions of the theorem.