INDIANAPOLIS — In Kurt Vonnegut’s novel “Cat’s Cradle,” the narrator meets a woman on a plane who is delighted to discover that he is from Indiana. Holding his arm tightly, she tells him, “We Hoosiers got to stick together.”

Mr. Vonnegut’s writing was filled with references to his Midwestern roots and to the tight-knit families he met growing up here. Still, some readers may be surprised that his memorial library is opening in his hometown, Indianapolis, and not on the East Coast, where he lived for most of his life.

As the library welcomed the public for the first time last week, the author’s friends and family said that it belonged in Indianapolis, with which he had a complicated and not always complimentary relationship. Despite his criticism of the traditionally conservative city, this is where he developed his voice as a writer and learned the values expressed in his books....