# Triangulation Definition Via Cell Partitions

There are two ways of defining a CW-complex. The first is to "inductively build" CW-complexes: you start with 0-cells as your 0-skeleton, attach 1-cells to that to get your 1-skeleton,... and so on. Then one may define a CW-complex to be a space $X$ along with a homeomorphism $h \colon X \rightarrow C$, where $C$ is one of the spaces constructed in the above fashion (and I suppose that one should also remember the construction of $C$, to be equivalent to the below).

An alternative, equivalent definition is the following. One says that a CW-complex is a space $X$ along with a partition of $X$ into open cells. This decomposition needs to satisfy the usual properties.

My question is on the situation for triangulated spaces. One may define (in the fashion of the first approach to CW-complexes) a triangulation of a space $X$ as a homeomorphism $h \colon X \rightarrow K$, where $K$ is some (let's call it) polyhedron. Some like to think of a polyheron as a space living in some $\mathbb{R}^n$, with a decomposition (satisfying the usual rules) into simplices. A better definition is to say that a polyhedron is the geometric realisation of an abstract simplicial complex.

But what about an analogue to the second approach for a CW-complex? That is, can we define a triangulation of a space as a decomposition into simplices, satisfying certain properties? The obvious approach, to me, would be to say that a triangulated space is a CW-complex (that is, it posseses a certain "nice" decomposition into cells), which satisfies the following:

Each $i$-cell $c$ is paired with a characteristic function $f_c \colon \Delta^i \rightarrow X$ (where $\Delta^i$ is the standard $i$-simplex). This data should satisfy the following:

1. Each $f_c$ is a homeomorphism onto its image.
2. The restriction of any $f_c$ to any face $F \subset \Delta^i$ is itself a characteristic function for some cell, perhaps after precomposing with some homeomorphism $h \colon \Delta^{i-1} \rightarrow F$.

This definition is slightly lacking, since I won't be able to do barycentric subdivision on this complex (for example, the barycentre of a simplex in $X$ may not correspond to its barycentre viewed as a face of a higher dimensional simplex). So I suppose that one could replace 2. with the condition that $h$ is linear.

Q: Would this definition be a suitable, instrinsic definition of a triangulated space, in terms of decomposing it into singular simplexes?

It is unnecessary to "replace 2. with the condition that $h$ is linear", because you can always recover that condition after the fact. Assuming that this condition holds for all characteristic maps of simplices of dimension $\le i-1$, given an arbitrary characteristic map $f_c \mid \Delta^i \to X^{i-1}$ satisfying conditions 1 and 2, you may precompose $f_c$ by a skeleton preserving isotopy of $\Delta^i$ so that its restriction to each face is linear. The reason this is possible is because of the theorem that two homeomorphisms from the $k$-ball $B^k$ to itself which are equal on the boundary $S^{k-1}$ are isotopic relative to $S^{k-1}$.
I should say, however, that you will not necessarily get a simplicial complex by this method, instead you will get what is called a "$\Delta$-complex" in Hatcher's textbook. In order to get a simplicial complex the additional information needed is that the intersection of any two simplices is another simplex (or empty).
• Many thanks for your answer. Yes, it is clear that I don't need $h$ linear, that's a good observation. As I remember, the take-home message of the difference between a simplicial complex and of a $\Delta$-complex is that in a simplicial complex you don't get any identifications between points of the same simplex. But isn't that covered by my condition 1. (I ask that the $f_c$'s are homeomorphisms)? Otherwise, my definition above looks very similar to that in Hatcher's book, although in a $\Delta$-complex one has an ordering on the vertices. – Jamie Walton Aug 20 '14 at 16:29
• Ahha, I see. I still need the intersection condition, whatever the thing I've defined is. The problem is that one could have, for example, two simplexes meeting along a subcomplex, instead of just a common face. So the above (perhaps pointless) definition sits somewhere in between $\Delta$-complexes and simplicial complexes (since in $\Delta$-complexes it is not required that the maps $f_c \colon \Delta^i \rightarrow X$ are injective). – Jamie Walton Aug 20 '14 at 17:21