Which layer of the mantle is soft and allows the lithosphere to float on it?

The asthenosphere is the correct choice because it is the layer of the upper mantle that is characterized by its semi-fluid or plastic-like properties. This unique characteristic allows the rigid lithosphere, which includes the crust and the uppermost portion of the mantle, to float and move atop it. The asthenosphere is composed of rock that is close to its melting point, allowing it to deform and flow slowly over geological timescales. This movement is crucial for tectonic processes such as plate tectonics, where the lithospheric plates move and interact due to convection currents within the asthenosphere.

The outer core, while important in generating Earth's magnetic field, is a liquid layer composed mainly of iron and nickel and does not have the same structural relationship with the lithosphere. The crust, which is the Earth’s outermost layer, is rigid and does not support the lithosphere's floating characteristic on its own. The lower mantle is solid and does not exhibit the soft, flowing behavior of the asthenosphere, making it unable to support the lithosphere.