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](https://www.ams.org/journals/tran/2009-361-11/S0002-9947-09-04898-3/S0002-9947-09-04898-3.pdf). 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.



Contractible subcomplex containing 1-skeleton?
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.


