diff --git a/Combinatorial_map/include/CGAL/Cell_attribute.h b/Combinatorial_map/include/CGAL/Cell_attribute.h
index 7718567dd26..d03695d6d0c 100644
--- a/Combinatorial_map/include/CGAL/Cell_attribute.h
+++ b/Combinatorial_map/include/CGAL/Cell_attribute.h
@@ -34,10 +34,10 @@ class Combinatorial_map_storage_1;
 template<unsigned int, class, class>
 class Combinatorial_map_storage_2;
 
-template<unsigned int, class, class, class>
+template<unsigned int, class, class>
 class Generalized_map_storage_1;
 
-template<unsigned int, class, class, class>
+template<unsigned int, class, class>
 class Generalized_map_storage_2;
 
 template<unsigned int, unsigned int, class, class, class, class>
@@ -113,10 +113,10 @@ struct Init_id;
     template<unsigned int, class, class>
     friend class Combinatorial_map_storage_2;
 
-    template<unsigned int, class, class, class>
+    template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
-    template<unsigned int, class, class, class>
+    template<unsigned int, class, class>
     friend class Generalized_map_storage_2;
 
     template<unsigned int, unsigned int, class, class, class, class>
@@ -238,10 +238,10 @@ struct Init_id;
     template<unsigned int, class, class>
     friend class Combinatorial_map_storage_2;
 
-    template<unsigned int, class, class, class>
+    template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
-    template<unsigned int, class, class, class>
+    template<unsigned int, class, class>
     friend class Generalized_map_storage_2;
 
     template<unsigned int, unsigned int, class, class, class, class>
@@ -425,10 +425,10 @@ struct Init_id;
     template<unsigned int, class, class>
     friend class Combinatorial_map_storage_2;
 
-    template<unsigned int, class, class, class>
+    template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
-    template<unsigned int, class, class, class>
+    template<unsigned int, class, class>
     friend class Generalized_map_storage_2;
 
     template<unsigned int, unsigned int, class, class, class, class>
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_storages_with_index.h b/Combinatorial_map/include/CGAL/Combinatorial_map_storages_with_index.h
index e20215fe0ea..ef52449df94 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_storages_with_index.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_storages_with_index.h
@@ -38,9 +38,11 @@ namespace CGAL {
   {
   public:
     using Self=Combinatorial_map_storage_2<d_, Items_, Alloc_>;
+    using Use_index=CGAL::Tag_true;
+    using Concurrent_tag=typename internal::Get_concurrent_tag<Items_>::type;
+
     typedef internal::Combinatorial_map_helper<Self> Helper;
 
-    using Use_index=CGAL::Tag_true;
     using Index_type=typename internal::Get_index_type<Items_>::type;
 
     typedef typename Items_::template Dart_wrapper<Self>  Dart_wrapper;
diff --git a/Combinatorial_map/include/CGAL/Dart.h b/Combinatorial_map/include/CGAL/Dart.h
index 4deb78f7dd0..dba2af5b116 100644
--- a/Combinatorial_map/include/CGAL/Dart.h
+++ b/Combinatorial_map/include/CGAL/Dart.h
@@ -35,10 +35,10 @@ namespace CGAL {
   template<unsigned int, class, class>
   class Combinatorial_map_storage_2;
 
-  template<unsigned int, class, class, class>
+  template<unsigned int, class, class>
   class Generalized_map_storage_1;
 
-  template<unsigned int, class, class, class>
+  template<unsigned int, class, class>
   class Generalized_map_storage_2;
 
   template<unsigned int, unsigned int, class, class, class, class>
@@ -90,10 +90,10 @@ namespace CGAL {
     template<unsigned int, class, class>
     friend class Combinatorial_map_storage_2;
 
-    template<unsigned int, class, class, class>
+    template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
-    template<unsigned int, class, class, class>
+    template<unsigned int, class, class>
     friend class Generalized_map_storage_2;
 
     template<unsigned int, unsigned int, class, class, class, class>
@@ -253,10 +253,10 @@ namespace CGAL {
     template<unsigned int, class, class>
     friend class Combinatorial_map_storage_2;
 
-    template<unsigned int, class, class, class>
+    template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
-    template<unsigned int, class, class, class>
+    template<unsigned int, class, class>
     friend class Generalized_map_storage_2;
 
     template<unsigned int, unsigned int, class, class, class, class>
diff --git a/Generalized_map/include/CGAL/Generalized_map.h b/Generalized_map/include/CGAL/Generalized_map.h
index f0b0f8a007e..4e148fea569 100644
--- a/Generalized_map/include/CGAL/Generalized_map.h
+++ b/Generalized_map/include/CGAL/Generalized_map.h
@@ -3607,7 +3607,7 @@ namespace CGAL {
       else
       {
         if (are_attributes_automatically_managed() &&
-            update_attributes && ah!=nullptr)
+            update_attributes && ah!=null_handle)
         {
           internal::Set_i_attribute_of_dart_functor<Self, 0>::run(*this, d2, ah);
           if (!isfree1)
@@ -3833,7 +3833,7 @@ namespace CGAL {
               basic_link_alpha(dd, dddd, dim);
               basic_link_alpha(alpha<0>(dd), d0, dim);
 
-              if (oldb2!=nullptr)
+              if (oldb2!=null_handle)
               {
                 basic_link_alpha<2>(alpha(oldb2, dim), dddd);
                 basic_link_alpha<2>(alpha(oldb2, 0, dim), d0);
@@ -3987,68 +3987,6 @@ namespace CGAL {
     {}
   };
 
-  namespace Index
-  {
-  template<unsigned int d_, class Items_, class Alloc_,class Storage_>
-  class Generalized_map :
-    public Generalized_map_base<d_,
-                                  Generalized_map<d_,Items_,Alloc_, Storage_>,
-                                  Items_, Alloc_, Storage_ >
-  {
-  public:
-    typedef Generalized_map<d_, Items_,Alloc_, Storage_>  Self;
-    typedef Generalized_map_base<d_, Self, Items_, Alloc_, Storage_> Base;
-
-    typedef typename Base::Dart_handle Dart_handle;
-    typedef typename Base::Dart_const_handle Dart_const_handle;
-    typedef typename Base::Alloc Alloc;
-    typedef typename Base::Exception_no_more_available_mark
-    Exception_no_more_available_mark;
-
-    Generalized_map() : Base()
-    {}
-
-    Generalized_map(const Self & amap) : Base(amap)
-    {}
-
-    Generalized_map(Self && amap) : Base(amap)
-    {}
-
-    template <unsigned int d2, typename Refs2, typename Items2, typename Alloc2,
-              typename Storage2>
-    Generalized_map(const Generalized_map_base<d2, Refs2, Items2, Alloc2, Storage2>& amap) :
-      Base(amap)
-    {}
-
-    template <unsigned int d2, typename Refs2, typename Items2, typename Alloc2,
-              typename Storage2, typename Converters>
-    Generalized_map(const Generalized_map_base<d2, Refs2, Items2, Alloc2, Storage2>& amap,
-                    const Converters& converters) :
-      Base(amap, converters)
-    {}
-
-    template <unsigned int d2, typename Refs2, typename Items2, typename Alloc2,
-              typename Storage2,
-              typename Converters, typename DartInfoConverter>
-    Generalized_map(const Generalized_map_base<d2, Refs2, Items2, Alloc2, Storage2>& amap,
-                    const Converters& converters,
-                    const DartInfoConverter& dartinfoconverter) :
-      Base(amap, converters, dartinfoconverter)
-    {}
-
-    template <unsigned int d2, typename Refs2, typename Items2, typename Alloc2,
-              typename Storage2,
-              typename Converters, typename DartInfoConverter,
-              typename PointConverter >
-    Generalized_map(const Generalized_map_base<d2, Refs2, Items2, Alloc2, Storage2>& amap,
-                    const Converters& converters,
-                    const DartInfoConverter& dartinfoconverter,
-                    const PointConverter& pointconverter) :
-      Base(amap, converters, dartinfoconverter, pointconverter)
-    {}
-  };
-  } // namespace Index
-
 } // namespace CGAL
 
 #if defined(BOOST_GCC)
diff --git a/Generalized_map/include/CGAL/Generalized_map_fwd.h b/Generalized_map/include/CGAL/Generalized_map_fwd.h
index 0ca8a8bd078..b73b1387d38 100644
--- a/Generalized_map/include/CGAL/Generalized_map_fwd.h
+++ b/Generalized_map/include/CGAL/Generalized_map_fwd.h
@@ -14,24 +14,60 @@
 
 #include <CGAL/memory.h>
 #include <CGAL/tags.h>
+#include <boost/mpl/has_xxx.hpp>
 
 namespace CGAL {
 
-template<unsigned int d_, class Items_, class Alloc_, class Concurrent_tag=CGAL::Tag_false >
+template<unsigned int d_, class Items_, class Alloc_>
 class Generalized_map_storage_1;
 
+template<unsigned int d_, class Items_, class Alloc_>
+class Generalized_map_storage_2;
+
 struct Generic_map_min_items;
 
-template < unsigned int d_, class Refs,
-           class Items_=Generic_map_min_items,
-           class Alloc_=CGAL_ALLOCATOR(int),
-           class Storage_= Generalized_map_storage_1<d_, Items_, Alloc_, CGAL::Tag_false> >
+namespace internal
+{
+template<typename Tag>
+struct Default_storage_for_gmap_when_tag
+{
+  template<unsigned int d_, class Items_, class Alloc_>
+  using type=Generalized_map_storage_1<d_, Items_, Alloc_>;
+};
+template<>
+struct Default_storage_for_gmap_when_tag<CGAL::Tag_true>
+{
+  template<unsigned int d_, class Items_, class Alloc_>
+  using type=Generalized_map_storage_2<d_, Items_, Alloc_>;
+};
+
+BOOST_MPL_HAS_XXX_TRAIT_NAMED_DEF(Has_use_index_tag,Use_index,false)
+template<typename T, bool typedefined=Has_use_index_tag<T>::value>
+struct Default_storage_for_gmap
+{
+  template<unsigned int d_, class Items_, class Alloc_>
+  using type=Generalized_map_storage_1<d_, Items_, Alloc_>;
+};
+template<typename T>
+struct Default_storage_for_gmap<T, true>
+{
+  template<unsigned int d_, class Items_, class Alloc_>
+  using type=typename Default_storage_for_gmap_when_tag<typename T::Use_index>::
+  type<d_, Items_, Alloc_>;
+};
+} // namespace internal
+template<unsigned int d_, class Refs_,
+         class Items_=Generic_map_min_items,
+         class Alloc_=CGAL_ALLOCATOR(int),
+         class Storage_=typename internal::Default_storage_for_gmap<Items_>::
+         type<d_, Items_, Alloc_>>
 class Generalized_map_base;
 
-template < unsigned int d_,
-           class Items_=Generic_map_min_items,
-           class Alloc_=CGAL_ALLOCATOR(int),
-           class Storage_= Generalized_map_storage_1<d_, Items_, Alloc_, CGAL::Tag_false> >
+template <unsigned int d_,
+          class Items_=Generic_map_min_items,
+          class Alloc_=CGAL_ALLOCATOR(int),
+         class Storage_=typename internal::Default_storage_for_gmap<Items_>::
+         type<d_, Items_, Alloc_>>
 class Generalized_map;
 
 } // CGAL
diff --git a/Generalized_map/include/CGAL/Generalized_map_storages.h b/Generalized_map/include/CGAL/Generalized_map_storages.h
index d28a050f42b..e8650e58fc8 100644
--- a/Generalized_map/include/CGAL/Generalized_map_storages.h
+++ b/Generalized_map/include/CGAL/Generalized_map_storages.h
@@ -35,12 +35,13 @@ namespace CGAL {
   }
 
   // Storage of darts with compact container, alpha with handles
-  template<unsigned int d_, class Items_, class Alloc_, class Concurrent_tag >
+  template<unsigned int d_, class Items_, class Alloc_>
   class Generalized_map_storage_1
   {
   public:
-    using Self=Generalized_map_storage_1<d_, Items_, Alloc_, Concurrent_tag>;
+    using Self=Generalized_map_storage_1<d_, Items_, Alloc_>;
     using Use_index=CGAL::Tag_false;
+    using Concurrent_tag=typename internal::Get_concurrent_tag<Items_>::type;
 
     typedef internal::Combinatorial_map_helper<Self>      Helper;
 
@@ -65,6 +66,8 @@ namespace CGAL {
     typedef std::nullptr_t Null_handle_type;
     static const Null_handle_type null_handle;
 
+    using Type_for_compact_container=void*;
+
     typedef Items_ Items;
     typedef Alloc_ Alloc;
     template <typename T>
@@ -433,9 +436,9 @@ namespace CGAL {
   };
 
   /// null_handle
-  template < unsigned int d_, class Items_, class Alloc_, class Concurrent_tag >
-  const typename Generalized_map_storage_1<d_, Items_, Alloc_, Concurrent_tag>::Null_handle_type
-  Generalized_map_storage_1<d_, Items_, Alloc_, Concurrent_tag>::null_handle = nullptr;
+  template < unsigned int d_, class Items_, class Alloc_>
+  const typename Generalized_map_storage_1<d_, Items_, Alloc_>::Null_handle_type
+  Generalized_map_storage_1<d_, Items_, Alloc_>::null_handle = nullptr;
 
 } // namespace CGAL
 
diff --git a/Generalized_map/include/CGAL/Generalized_map_storages_with_index.h b/Generalized_map/include/CGAL/Generalized_map_storages_with_index.h
index e9ba38f36ed..7fc51f9f651 100644
--- a/Generalized_map/include/CGAL/Generalized_map_storages_with_index.h
+++ b/Generalized_map/include/CGAL/Generalized_map_storages_with_index.h
@@ -33,21 +33,23 @@ namespace CGAL {
   }
 
   // Storage of darts with compact container, alpha with indices
-  template<unsigned int d_, class Items_, class Alloc_, class Index_type_ >
+  template<unsigned int d_, class Items_, class Alloc_>
   class Generalized_map_storage_2
   {
   public:
-    using Self=Generalized_map_storage_2<d_, Items_, Alloc_, Index_type_>;
+    using Self=Generalized_map_storage_2<d_, Items_, Alloc_>;
     using Use_index=CGAL::Tag_true;
-    using Index_type=Index_type_;
+    using Concurrent_tag=typename internal::Get_concurrent_tag<Items_>::type;
 
     typedef internal::Combinatorial_map_helper<Self> Helper;
 
+    using Index_type=typename internal::Get_index_type<Items_>::type;
+
     typedef typename Items_::template Dart_wrapper<Self>  Dart_wrapper;
 
     typedef typename internal::template Get_dart_info<Dart_wrapper>::type
                                                            Dart_info;
-    typedef CGAL::Index::Dart<d_, Self, Dart_info> Dart;
+    typedef CGAL::Dart<d_, Self, Dart_info> Dart;
 
     typedef std::allocator_traits<Alloc_> Allocator_traits;
     typedef typename Allocator_traits::template rebind_alloc<Dart> Dart_allocator;
@@ -68,6 +70,8 @@ namespace CGAL {
     typedef Dart_index Null_handle_type;
     static Null_handle_type null_handle;
 
+    using Type_for_compact_container=Index_type;
+
     typedef Items_ Items;
     typedef Alloc_ Alloc;
     template <typename T>
@@ -114,7 +118,7 @@ namespace CGAL {
     
     // Init
     void init_storage()
-    { null_dart_handle=nullptr; }
+    {}
 
     void clear_storage()
     {}
@@ -388,13 +392,13 @@ namespace CGAL {
     void dart_link_alpha(Dart_handle adart, Dart_handle adart2)
     {
       CGAL_assertion(i <= dimension);
-      CGAL_assertion(adart!=nullptr && adart2!=nullptr);
+      CGAL_assertion(adart!=null_handle && adart2!=null_handle);
       mdarts[adart].mf[i] = adart2;
     }
     void dart_link_alpha(Dart_handle adart, Dart_handle adart2, unsigned int i)
     {
       CGAL_assertion(i <= dimension);
-      CGAL_assertion(adart!=nullptr && adart2!=nullptr);
+      CGAL_assertion(adart!=null_handle && adart2!=null_handle);
       mdarts[adart].mf[i] = adart2;
     }
 
@@ -405,12 +409,12 @@ namespace CGAL {
     template<unsigned int i>
     void dart_unlink_alpha(Dart_handle adart)
     {
-      CGAL_assertion(adart!=nullptr && i <= dimension);
+      CGAL_assertion(adart!=null_handle && i <= dimension);
       mdarts[adart].mf[i] = adart;
     }
     void dart_unlink_alpha(Dart_handle adart, unsigned int i)
     {
-      CGAL_assertion(adart!=nullptr && i <= dimension);
+      CGAL_assertion(adart!=null_handle && i <= dimension);
       mdarts[adart].mf[i] = adart;
     }
 
@@ -425,9 +429,9 @@ namespace CGAL {
   };
 
   /// null_handle
-  template<unsigned int d_, class Items_, class Alloc_, class Size_type>
-  typename Generalized_map_storage_2<d_, Items_, Alloc_, Size_type>::Null_handle_type
-      Generalized_map_storage_2<d_, Items_, Alloc_, Size_type>::null_handle((std::numeric_limits<size_type>::max)()/2);
+  template<unsigned int d_, class Items_, class Alloc_>
+  typename Generalized_map_storage_2<d_, Items_, Alloc_>::Null_handle_type
+      Generalized_map_storage_2<d_, Items_, Alloc_>::null_handle((std::numeric_limits<size_type>::max)()/2);
 
 } // namespace CGAL
 
diff --git a/Generalized_map/test/Generalized_map/CMakeLists.txt b/Generalized_map/test/Generalized_map/CMakeLists.txt
index fec14337999..8d6c93723c4 100644
--- a/Generalized_map/test/Generalized_map/CMakeLists.txt
+++ b/Generalized_map/test/Generalized_map/CMakeLists.txt
@@ -19,11 +19,12 @@ if(NOT Boost_FOUND)
 
 endif()
 
-# include for local directory
+set(hfiles Generalized_map_2_test.h Generalized_map_3_test.h 
+  Generalized_map_4_test.h GMap_test_insertions.h)
 
-# include for local package
+create_single_source_cgal_program("Generalized_map_test.cpp" ${hfiles})
 
-# Creating entries for all C++ files with "main" routine
-# ##########################################################
+add_executable(Generalized_map_test_index  Generalized_map_test.cpp ${hfiles})
+target_compile_definitions(Generalized_map_test_index PUBLIC USE_COMPACT_CONTAINER_WITH_INDEX)
+target_link_libraries(Generalized_map_test_index PUBLIC CGAL CGAL::Data)
 
-create_single_source_cgal_program("Generalized_map_test.cpp")
diff --git a/Generalized_map/test/Generalized_map/Generalized_map_test.cpp b/Generalized_map/test/Generalized_map/Generalized_map_test.cpp
index ca12ca9b7e7..ae7f2149e68 100644
--- a/Generalized_map/test/Generalized_map/Generalized_map_test.cpp
+++ b/Generalized_map/test/Generalized_map/Generalized_map_test.cpp
@@ -21,8 +21,18 @@ struct Divide_by_two_functor
   }
 };
 
+struct Min_items: public CGAL::Generic_map_min_items
+{
+#ifdef USE_COMPACT_CONTAINER_WITH_INDEX
+  typedef CGAL::Tag_true Use_index;
+#endif
+};
+
 struct Myitems_2
 {
+#ifdef USE_COMPACT_CONTAINER_WITH_INDEX
+  typedef CGAL::Tag_true Use_index;
+#endif
   template <class GMAP>
   struct Dart_wrapper
   {
@@ -37,6 +47,9 @@ struct Myitems_2
 
 struct Myitems_2c
 {
+#ifdef USE_COMPACT_CONTAINER_WITH_INDEX
+  typedef CGAL::Tag_true Use_index;
+#endif
   template <class GMAP>
   struct Dart_wrapper
   {
@@ -50,6 +63,9 @@ struct Myitems_2c
 };
 struct Myitems_3
 {
+#ifdef USE_COMPACT_CONTAINER_WITH_INDEX
+  typedef CGAL::Tag_true Use_index;
+#endif
   template <class GMAP>
   struct Dart_wrapper
   {
@@ -64,6 +80,9 @@ struct Myitems_3
 
 struct Myitems_3c
 {
+#ifdef USE_COMPACT_CONTAINER_WITH_INDEX
+  typedef CGAL::Tag_true Use_index;
+#endif
   template <class GMAP>
   struct Dart_wrapper
   {
@@ -78,6 +97,9 @@ struct Myitems_3c
 
 struct Myitems_4
 {
+#ifdef USE_COMPACT_CONTAINER_WITH_INDEX
+  typedef CGAL::Tag_true Use_index;
+#endif
   template <class GMAP>
   struct Dart_wrapper
   {
@@ -92,6 +114,9 @@ struct Myitems_4
 
 struct Myitems_4c
 {
+#ifdef USE_COMPACT_CONTAINER_WITH_INDEX
+  typedef CGAL::Tag_true Use_index;
+#endif
   template <class GMAP>
   struct Dart_wrapper
   {
@@ -109,7 +134,7 @@ int main()
   std::cout<<"Generalized_map start test (v1)."<<std::flush;
 
   trace_display_msg("\ntest_GMAP_2<GMAP2>");
-  typedef CGAL::Generalized_map<2> GMAP2;
+  typedef CGAL::Generalized_map<2, Min_items> GMAP2;
   if ( !test_GMAP_2<GMAP2>() )
   {
     std::cout<<" Error during Test_GMAP_2<GMAP2>."<<std::endl;
@@ -117,7 +142,7 @@ int main()
   }
 
   trace_display_msg("test_GMAP_3<GMAP3>");
-  typedef CGAL::Generalized_map<3> GMAP3;
+  typedef CGAL::Generalized_map<3, Min_items> GMAP3;
   if ( !test_GMAP_3<GMAP3>() )
   {
     std::cout<<" Error during Test_GMAP_3<GMAP3>."<<std::endl;
@@ -125,7 +150,7 @@ int main()
   }
 
   trace_display_msg("test_GMAP_4<GMAP4>");
-  typedef CGAL::Generalized_map<4> GMAP4;
+  typedef CGAL::Generalized_map<4, Min_items> GMAP4;
   if ( !test_GMAP_4<GMAP4>() )
   {
     std::cout<<" Error during Test_GMAP_4<GMAP4>."<<std::endl;