  Interactions in complex systems are widely observed across various fields,
drawing increased attention from researchers. In mathematics, efforts are made
to develop various theories and methods for studying the interactions between
spaces. In this work, we present an algebraic topology framework to explore
interactions between spaces. We introduce the concept of interaction spaces and
investigate their homotopy, singular homology, and simplicial homology.
Furthermore, we demonstrate that interaction singular homology serves as an
invariant under interaction homotopy. We believe that the proposed framework
holds potential for practical applications.
