Do isomorphic simplicial chain complexes give isomorphic simplicial complexes?

Question

It seems that chain complexes have all information of simplicial complexes. If we have an isomorphism between chain complexes which are induced by simplicial complexes, can we conclude that two simplicial complexes are isomorphic?

Answer 1

No. For instance, consider two non-isomorphic trees with the same number of vertices. Both will have a chain complex of the form $0\to\mathbb{Z}^{n-1}\to\mathbb{Z}^n\to 0$ where the map $\mathbb{Z}^{n-1}\to\mathbb{Z}^n$ is injective and its cokernel is free of rank $1$. Any two such chain complexes are isomorphic, since the images of the standard basis elements of $\mathbb{Z}^{n-1}$ together with a lift of a generator of the cokernel will form a basis for $\mathbb{Z}^n$, so you can choose a basis for $\mathbb{Z}^n$ such that the map $\mathbb{Z}^{n-1}\to\mathbb{Z}^n$ is just the inclusion given by the first $n-1$ basis vectors.