Contractible subcomplex containing 1-skeleton? Question: If $X$ is a simplicial complex that's simply connected and $2$-dimensional, does there always exist a contractible subcomplex $Y$ satisfying $X^{(1)} \subseteq Y$?
The statement is true "down a dimension": If $X$ is connected and $1$-dimensional, then there exists a contractible subcomplex $Y$ satisfying $X^{(0)} \subseteq Y$, namely you can use a spanning tree. So, one could think of the desired $Y$ in the original problem as a $2$-dimensional analog of a spanning tree, with contractibility being the key desired property.
It seems there is a higher dimensional analog of "spanning tree" in the literature, at least for finite complexes, e.g., Definition 3.1 of this. Note that Proposition 3.7 of that paper implies that (finite) simply connected $2$-complexes indeed have these sorts of $2$-dimensional "spanning trees". But these are not necessarily contractible, they're more of a homological analog of trees.
 A: I think the answer is no.
In Hatcher's Algebraic Topology there is an example of an acyclic 2-dimensional complex with one 0-cell, two 1-cells and two 2-cells (Example 2.38). You start with a wedge of two circles, labeled $a$ and $b$, and attach two 2-dimensional cells by the words $a^5b^{-3}$ and $b^2(ab)^{-2}$. One can easily check that it is acyclic. The fundamental group has the presentation $<a, b\mid a^5b^{-3}, b^2(ab)^{-2}>$, and one can show that it is not trivial. Now form a new complex $X$ by attaching one more $2$-dimensional cell, along $b$. The fundamental group of $X$ is easily seen to be trivial. But no proper subcomplex of $X$ containing the $1$-skeleton is simply-connected, let alone contractible.
To make this example into a simplicial complex, choose a triangulation of $X$. This can be done because all the attaching maps are nice piece-wise linear maps. I believe it is still true that no simplicial subcomplex of $X$ that contains the $1$-skeleton is contractible. Removing a two-cell from the triangulation of $X$ is like punching a hole in a two cell in the original CW structure on $X$. We already know that punching a single hole in some of the cells does not make a contractible complex, and punching additional holes only makes the fundamental group bigger.
