How can I use EF6 to update a many to many table

I think you are trying to simulating offline mode working for your users. So when you get something from your users, you want to sync database with user data. I make an example and take your question one step beyond :) I added a Subtopic which needs to be updated in database. Ok here is the code:

static void Main(string[] args)
{
    //the database
    var ObjectiveDetails = new List<ObjectiveDetail>()
    {
        new ObjectiveDetail()
        {
            ObjectiveDetailId = 1,
            Number = 1,
            Text = "datafromdb",
            SubTopics = new List<SubTopic>()
            {
                new SubTopic(){ SubTopicId = 1, Name="one"}, //no change
                new SubTopic(){ SubTopicId = 2, Name="two"}, //to be deleted
                new SubTopic(){ SubTopicId = 4, Name="four"} //to be updated
            }
        }
    };

    //the object comes as json and serialized to defined object.
    var web = new ObjectiveDetail()
    {
        ObjectiveDetailId = 1,
        Number = 1,
        Text = "datafromweb",
        SubTopics = new List<SubTopic>()
        {
            new SubTopic(){ SubTopicId = 1, Name="one"}, //no change
            new SubTopic(){ SubTopicId = 3, Name="three"}, //new row
            new SubTopic(){ SubTopicId = 4, Name="new four"} //must be updated
        }
    };

    var objDet = ObjectiveDetails.FirstOrDefault(x => x.ObjectiveDetailId == web.ObjectiveDetailId);
    if (objDet != null)
    {
        //you can use AutoMapper or ValueInjecter for mapping and binding same objects
        //but it is out of scope of this question
        //update ObjectDetail
        objDet.Number = web.Number;
        objDet.Text = web.Text;
        var subtops = objDet.SubTopics.ToList();

        //Delete removed parameters from database
        //Entity framework can handle it for you via change tracking
        //subtopicId = 2 has been deleted 
        subtops.RemoveAll(x => !web.SubTopics.Select(y => y.SubTopicId).Contains(x.SubTopicId));

        //adds new items which comes from web
        //adds subtopicId = 3 to the list
        var newItems = web.SubTopics.Where(x => !subtops.Select(y => y.SubTopicId).Contains(x.SubTopicId)).ToList();
        subtops.AddRange(newItems);

        //this items must be updated
        var updatedItems = web.SubTopics.Except(newItems).ToList();

        foreach (var item in updatedItems)
        {
            var dbItem = subtops.First(x => x.SubTopicId == item.SubTopicId);
            dbItem.Name = item.Name;
        }

        //let's see is it working
        Console.WriteLine("{0}:\t{1}\t{2}\n---------",objDet.ObjectiveDetailId, objDet.Number, objDet.Text);
        foreach (var item in subtops)
        {
            Console.WriteLine("{0}: {1}", item.SubTopicId, item.Name);
        }
    }
    else
    {
         //insert new ObjectiveDetail
    }

    //In real scenario after doing everything you need to call SaveChanges or it's equal in your Unit of Work.
}

The result:

1:      1       datafromweb
---------
1: one
4: new four
3: three

That's it. You can sync your database and user data like this. And also AutoMapper and ValueInjecter both are very useful and powerful tools, I deeply recommend you to take a look at those. I hope you enjoyed, happy coding :)