By facilitating multiple independent owners to jointly control a distributed network, blockchain can be used solve the problem of device collaboration in complex networks (e.g., 5G, health care industries) through consensus mechanism. However, state-of-the-art blockchain-based solutions cannot meet demand high transaction rate for those applications, due unavoidable data synchronization cost de...

We write down the chain complex C∗ of the universal cover of the torus, and indicate how to do likewise for any 2-dimensional CW complex arising as the 2-skeleton of a K(G, 1). We exhibit a contracting chain homotopy for this chain complex, and again explain how to do so for the chain complex of the universal cover of our more general CW complex. This yields a contracting homotopy for the tenso...