Non-contractible simplicial complex and Euler characteristic 1

Question

It is a specialisation of Is there a non-contractible Simplicial Complex with Euler Characteristic 1?

Here is a list of non-contractible simplicial complex with Euler characteristic $1$:

• non-homogeneous: an octahedron with an edge running through the middle (see this answer)
• non-oriented: triangulation of the real projective plane (see this paper)
• non-connected: triangulation of the disjoint union of a disk and a torus

Question: Is there a non-contractible homogeneous oriented connected simplicial complex with Euler Characteristic $1$?

Answer 1

$(S^1 \times S^3) \#\Bbb CP^2$

Answer 2

Let $X$ be a $2$-disk with two smaller open $2$-disks removed.

Then $\chi(X)=-1$ and therefore $\chi (X\times X)=(-1)^2 =1$.

Right?