This is an additional exercise of Hatcher's Algebraic Topology book (link: https://pi.math.cornell.edu/~hatcher/AT/AT-exercises.pdf, section 2.1).
Let $X$ be the standard $n$-simplex with its natural $\Delta$-complex structure and $A$ a subcomplex with $H_{n-1}$ nonzero. How can we show that $A=\partial X$? Using the simplical chain complex of $A$, we see that $A$ must be $(n-1)$-dimensional. There are exactly $n$ $(n-1)$-simplices in $X$. How can we show that $A$ must contain these all? It will suffice to show that if $A$ is the union of $<(n-1)$ faces of $X$ then $H_{n-1}(A)=0$. Any hints?
Look at the boundary map from degree $n$ to degree $n-1$, and use the fact that $H_{n-1}(X)=0$.