A clinical examination is necessary for appropriate diagnosis and management of your problem. Cracked skin on thumbs could be due to extremely cold weather or lack of moisture to the skin. Another possibility is that it could be due to irritation to the skin from the use of certain lotions and hand soaps. Eczema is a form of chronic dermatitis (rash).Allergic reaction is one of the important causes of eczema. Psoriasis is a long-term (chronic) skin problem that causes skin cells to grow too quickly, resulting in thick, white, silvery, or red patches of skin. It is useful to apply a hand cream that offers barrier protection and moisturizing effect. Try to limit exposure of cold and wind to the hand. For exact diagnosis and treatment see a dermatologist. Hope it helps.
Best luck and regards!