Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds

We construct an obstruction for the existence of embeddings of homology $3$-sphere into homology $S^3\times S^1$ under some cohomological condition. The obstruction is defined as an element in the filtered version of the instanton Floer cohomology due to R.Fintushel-R.Stern. We make use of the $\mathbb{Z}$-fold covering space of homology $S^3\times S^1$ and the instantons on it.