What are three of the significances of the Sun Dance in The Way to Rainy Mountain?
Although I am a bit confused as to what you mean by the word "significances," I can certainly tell you at least three things about the importance of the Sun Dance to the Kiowa tribe of Native Americans. The Sun Dance was the yearly ceremony where the Kiowa tribe members would worship both the Sun and the mysterious effigy of Tai-me.
Tai-me is a strange creature (neither fully bird nor fully mammal) that came to the Kiowa after being announced by the Crow. Tai-me supposedly came directly from the Sun, and the Kiowa began to worship this creature. Eventually, Tai-me returned to the sun, but left an effigy of itself for the Kiowa to worship. This effigy is wrapped in cloth and only revealed on one day a year: the day of the Sun Dance. On this day, the sacred bundle is opened and the Tai-me effigy is placed upon a pole in the middle of the Sun Dance lodge so every member of the tribe can see it. It is at this point that the worship begins.
There are other parts of The Way to Rainy Mountain that deal with the history of the Sun Dance. There are two dates that are quite significant: 1887 and 1890. In 1887, Momaday's own grandmother (Aho) was seven years old. It was on this date that Aho witnessed the last complete Sun Dance where all of the sacred rituals were completed. Another important date is 1890 because that was the date of the very last Sun Dance that was left incomplete. Again, Aho was a direct eyewitness. Why was this Sun Dance left incomplete? In the middle of the Sun Dance, the soldiers from Fort Sill arrived and dispersed the Kiowa tribe.