The unoriented bordism theory $MO$ has a map to $H\mathbb{F}_2$ which is easily described for a space $X$ by pushing forward the fundamental class of a singular manifold to $H_*(X)$. Since $MO$ and $H\mathbb{F}_2$ both factor through chain complexes, it is tempting to ask if this can be realized as a map of chain complexes. I believe this is unlikely because although all smooth manifolds admit a triangulation, this obviously cannot be done naturally. However, this makes me wonder about the following:

For a space $X$, let $D_n(X)$ denote the set of pairs $(M,f)$ where $M$ is a smooth manifold with a specified triangulation and $f$ is a map from $M$ into $X$. This is a chain complex by letting the boundary map take a manifold to its boundary. Denote the nth homology of this chain complex by $MT_n(X)$ (meant to stand for triangulated bordism). Alternatively, I believe we can describe $D_n(X)$ as tuples $(M,f,\sigma)$ where $(M,f)$ is a singular manifold and $\sigma$ is a cycle lifting the fundamental class $[M]$ subject to a few conditions.

Recall that one can describe $H\mathbb{F}_2(X)$ as the homology of the chain complex $C(X)$ where $C_n(X)$ is the set of pairs $(S,f)$ where $S$ is a simplicial complex that is the union of its n-simplices and $f$ is a map from $S$ into $X$. The boundary operator takes $S$ to the union of n-1 simplices that are incident to an odd number of n-simplices.

There is a chain map $i:D(X) \rightarrow C(X)$ given by forgetting the fact that the domain of $f$ is a manifold. As well, we may also forget the triangulation to get a map $j$ from $D(X)$ to the chain complex of singular manifolds. Since every smooth manifold has a triangulation, $j$ (hence $j_*$) is surjective. This implies $i_*$ is surjective since it factors through $j_*$ and the map $MO \rightarrow H\mathbb{F}_2$.

(1) Is $MT$ a homology theory?

(2) Supposing (1), is $MT$ a wedge of $H\mathbb{F}_2$?

(3) Supposing (1), is $MT$ a pullback of $H\mathbb{F}_2$ and $MO$?

because this is the definition of bordism groups.$\endgroup$ – Mike Miller Dec 31 '19 at 12:19