BottomNavigationView切换Fragment失效问题求助

我最近在做Android项目，想通过点击BottomNavigationView的Tab切换Fragment，但始终没效果。翻了好多视频教程都解决不了，完全搞不清自己代码哪里出问题了。而且我尝试用getSupportFragmentManager()的时候，代码直接标红报错了。下面是我的相关代码、布局文件，还有错误情况，恳请帮忙排查！

Java代码（MainActivity） #

package com.example.whatsoever.ttravel;

import android.app.Activity;
import android.os.Bundle;
import android.support.annotation.NonNull;
import android.support.design.widget.BottomNavigationView;
import android.view.MenuItem;
//import android.support.v4.app.Fragment;
//import android.support.v4.app.FragmentManager;

public class MainActivity extends Activity {
    private BottomNavigationView.OnNavigationItemSelectedListener mOnNavigationItemSelectedListener = 
        new BottomNavigationView.OnNavigationItemSelectedListener() {
        @Override
        public boolean onNavigationItemSelected(@NonNull MenuItem item) {
            switch (item.getItemId()) {
                case R.id.navigation_home:
                    category_map map = new category_map();
                    FragmentManager fragmentManager= getFragmentManager();
                    fragmentManager.beginTransaction(R.id.a,map,map.getTag()).commit();
                    return true;
                case R.id.navigation_dashboard:
                    return true;
            }
            return false;
        }
    };

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
        BottomNavigationView navigation = (BottomNavigationView) findViewById(R.id.navigation);
        navigation.setOnNavigationItemSelectedListener(mOnNavigationItemSelectedListener);
    }
}

布局文件（activity_main.xml） #

<?xml version="1.0" encoding="utf-8"?>
<android.support.constraint.ConstraintLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    xmlns:tools="http://schemas.android.com/tools"
    android:id="@+id/container"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    tools:context=".MainActivity">

    <FrameLayout
        android:id="@+id/a"
        android:layout_width="match_parent"
        android:layout_height="match_parent">
    </FrameLayout>

    <android.support.design.widget.BottomNavigationView
        android:id="@+id/navigation"
        android:layout_width="0dp"
        android:layout_height="wrap_content"
        android:layout_marginEnd="0dp"
        android:layout_marginStart="0dp"
        android:background="?android:attr/windowBackground"
        app:layout_constraintBottom_toBottomOf="parent"
        app:layout_constraintLeft_toLeftOf="parent"
        app:layout_constraintRight_toRightOf="parent"
        app:menu="@menu/navigation" />
</android.support.constraint.ConstraintLayout>

错误情况 #

报错显示无法解析getSupportFragmentManager()方法，对应代码标红。

内容的提问来源于stack exchange，提问作者MoTahir