How provide Up navigation with toolbar's home button on v7 toolbar

Did you have the below code to set your toolbar as the default activity actionbar?

setSupportActionBar(toolbar);

You did not set an image icon to the home button, maybe it shows up, but you just can't see it.

getSupportActionBar().setHomeAsUpIndicator(iconDrawable);
getSupportActionBar().setDisplayHomeAsUpEnabled(true);

and as @Gonzalo said, you also have to override the menu select event to handle the home button onClick event

@Override
public boolean onOptionsItemSelected(MenuItem item) {
    switch (item.getItemId()) {
    // Respond to the action bar's Up/Home button
    case android.R.id.home:
        //handle the home button onClick event here.
        return true;
    case android.R.id.other_menu
        return true
    }

    return super.onOptionsItemSelected(item);
}

and why didn't you have a appbarlayout to contain the toolbar?

<android.support.design.widget.AppBarLayout
        android:id="@+id/baseAppbarLayout"
        android:layout_width="match_parent"
        android:layout_height="wrap_content">

        <android.support.v7.widget.Toolbar
            android:id="@+id/baseToolbar"
            android:layout_width="match_parent"
            android:layout_height="?attr/actionBarSize"
            app:layout_scrollFlags="scroll|enterAlways"
            app:popupTheme="@style/ThemeOverlay.AppCompat.Light" />

</android.support.design.widget.AppBarLayout>

More actionbar implement details please have a look at my github project, hope it can help:

java code https://github.com/DanielShum/MaterialAppBase/blob/master/materialAppBaseLibrary/src/main/java/com/daililol/material/appbase/base/BaseActionbarActivity.java

xml code https://github.com/DanielShum/MaterialAppBase/blob/master/materialAppBaseLibrary/src/main/res/layout/base_actionbar_activity.xml

1- set your toolbar

Toolbar toolbar = findViewById(R.id.toolbar);

2- setup your icon

    if (toolbar != null) {
        setSupportActionBar(toolbar);
        getSupportActionBar().setDisplayHomeAsUpEnabled(true);
        getSupportActionBar().setHomeButtonEnabled(true);

    }

3- override this method

@Override
public boolean onSupportNavigateUp() {
    onBackPressed();
    return true;
}

It's very simple. Just do the following steps:

1. Add the toolbar to the activity_child.xml:

(In my project, this AppBarLayout is placed inside a ConstraintLayout)

<android.support.design.widget.AppBarLayout
            android:id="@+id/appBarLayout"
            android:layout_width="match_parent"
            android:layout_height="wrap_content"
            android:theme="@style/AppTheme.AppBarOverlay"
            app:layout_constraintEnd_toEndOf="parent"
            app:layout_constraintStart_toStartOf="parent"
            app:layout_constraintTop_toTopOf="parent">

            <android.support.v7.widget.Toolbar
                android:id="@+id/toolbar"
                android:layout_width="match_parent"
                android:layout_height="?attr/actionBarSize"
                android:background="?attr/colorPrimary"
                app:popupTheme="@style/AppTheme.PopupOverlay" />

</android.support.design.widget.AppBarLayout>

...

2. Set up the toolbar in the ChildActivity.java

public class ChildActivity extends AppCompatActivity
{

    @Override
    protected void onCreate(@Nullable Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_child);

        Toolbar toolbar = findViewById(R.id.toolbar);
        setSupportActionBar(toolbar);
        getSupportActionBar().setHomeAsUpIndicator(R.drawable.ic_arrow_back_black_24dp);
        getSupportActionBar().setDisplayHomeAsUpEnabled(true);
  }

  ...
}

3. set the parentActivityName property in AndroidManifest.xml

<activity android:name=".ChildActivity"
            android:theme="@style/AppTheme.NoActionBar"
            android:parentActivityName=".ParentActivity"/>

Important

For this to work properly, do not call finish() in the parent activity after you call startActivity(intent).