Let us evaluate all the statements:

• Starch is a polymer of $\alpha$-glucose: This is a correct statement. Starch is a storage polysaccharide in plants and is a polymer composed of $\alpha$-D-glucose units. It consists of two components: amylose (a linear polymer) and amylopectin (a branched polymer).
• Amylose is a component of cellulose: This is an incorrect statement. Amylose is a component of starch, not cellulose. Cellulose is a linear polymer of $\beta$-D-glucose.
• Proteins are composed of only one type of amino acid: This is an incorrect statement. Proteins are heteropolymers made up of 20 different types of $\alpha$-amino acids, not homopolymers of a single amino acid.
• In the cyclic structure of fructose, there are four carbons and one oxygen atom: This statement is incorrect as phrased. While the furanose ring of fructose contains four carbon atoms and one oxygen atom, the entire fructose molecule still has six carbon atoms ($\text{C}_6\text{H}_{12}\text{O}_6$).