I would like to use only Cognito User Pool, and therefore I want to use identity federation with Cognito User Pools, without Cognito Federated Identities (identity pools).>
it is possible to have a user pool with google configured as an identity provider without using Cognito Federated Identities. Cognito has SDKs available for Android, iOS, and Javascript, you can find them on github(https://github.com/aws/). Can you be more specific about the problem you are running into when trying to do this?