You might not relate the two together but strength in regards to your core can have a direct impact on your breathing and vice versa. I do recall a study on endurance athletes from 2007 that found by fatiguing the core during racing affected performance by causing more labored breathing. The diaphragm, which is the largest respiratory muscle act as both a stabilizing and respiratory muscle. If your core is weak you'll overtax the diaphragm forcing it pick up some of the slack affecting breathing efficiency or simply making breathing a llittle more difficult. So by strengthening the core you can actually affect a performance gain by breathing more efficiently.
A strong stable core provides more than just breathing performance, during cycling it helps a cyclist apply power from the center of the body providing a strong pillar so the athlete doesn’t have to rely primarily on the quads. It gives a stable platform or launch pad from which the quads can apply power more directly. When a cyclist's core is under conditioned and becomes fatigued they start a lateral rock that can affect the movement of the knees that can lead to injury. This rocking or twisting can also lead to back pain or discomfort as well as a loss of wattage. Toward the end of a long event when fatigue sets in is when all this stability work really pays off.