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 ef52449df94..c5b07dcac33 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_storages_with_index.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_storages_with_index.h
@@ -58,8 +58,6 @@ namespace CGAL {
     Multiply_by_two_policy_for_cc_with_size<64>, Index_type>
     Dart_container;
 
-    // typedef unsigned int Dart_index;
-    // typedef MyIndex<unsigned int> Dart_index;
     typedef typename Dart_container::Index Dart_index;
 
     // Definition of old types, for backward compatibility.
@@ -480,7 +478,7 @@ namespace CGAL {
   /// null_handle
   template<unsigned int d_, class Items_, class Alloc_>
   typename Combinatorial_map_storage_2<d_, Items_, Alloc_>::Null_handle_type
-      Combinatorial_map_storage_2<d_, Items_, Alloc_>::null_handle((std::numeric_limits<size_type>::max)()/2);
+      Combinatorial_map_storage_2<d_, Items_, Alloc_>::null_handle((std::numeric_limits<Index_type>::max)()/2);
 
 } // 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 7fc51f9f651..38b4a483179 100644
--- a/Generalized_map/include/CGAL/Generalized_map_storages_with_index.h
+++ b/Generalized_map/include/CGAL/Generalized_map_storages_with_index.h
@@ -58,8 +58,6 @@ namespace CGAL {
     Multiply_by_two_policy_for_cc_with_size<64>, Index_type>
     Dart_container;
 
-    // typedef unsigned int Dart_index;
-    // typedef MyIndex<unsigned int> Dart_index;
     typedef typename Dart_container::Index Dart_index;
 
     // Definition of old types, for backward compatibility.
@@ -431,7 +429,7 @@ namespace CGAL {
   /// null_handle
   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);
+      Generalized_map_storage_2<d_, Items_, Alloc_>::null_handle((std::numeric_limits<Index_type>::max)()/2);
 
 } // namespace CGAL
 
diff --git a/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages.h b/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages.h
index 38d000f9d06..78909b5abbc 100644
--- a/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages.h
+++ b/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages.h
@@ -40,13 +40,14 @@ namespace CGAL {
   // as template parameter of Dart_wrapper. If we inherit, Self is not
   // the correct type).
   template<unsigned int d_, unsigned int ambient_dim,
-           class Traits_, class Items_, class Alloc_, class Concurrent_tag >
+           class Traits_, class Items_, class Alloc_>
   class CMap_linear_cell_complex_storage_1
   {
   public:
     using Self=CMap_linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
-    Items_, Alloc_, Concurrent_tag>;
+    Items_, Alloc_>;
     using Use_index=CGAL::Tag_false;
+    using Concurrent_tag=typename internal::Get_concurrent_tag<Items_>::type;
 
     typedef typename Traits_::Point  Point;
     typedef typename Traits_::Vector Vector;
@@ -75,6 +76,8 @@ 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>
@@ -481,9 +484,9 @@ namespace CGAL {
   template<unsigned int d_, unsigned int ambient_dim,
            class Traits_, class Items_, class Alloc_, class Concurrent_tag >
   const typename CMap_linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
-                                         Items_, Alloc_, Concurrent_tag>::Null_handle_type
+                                         Items_, Alloc_>::Null_handle_type
   CMap_linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
-                                Items_, Alloc_, Concurrent_tag>::null_handle = nullptr;
+                                Items_, Alloc_>::null_handle = nullptr;
 
 } // namespace CGAL
 
diff --git a/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages_with_index.h b/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages_with_index.h
index 403195a9d2f..574915140b6 100644
--- a/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages_with_index.h
+++ b/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages_with_index.h
@@ -38,26 +38,24 @@ namespace CGAL {
   // as template parameter of Dart_wrapper. If we inherit, Self is not
   // the correct type).
   template<unsigned int d_, unsigned int ambient_dim, class Traits_, 
-           class Items_, class Alloc_, class Index_type_ >
+           class Items_, class Alloc_>
   class CMap_linear_cell_complex_storage_2
   {
   public:
     using Self=CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
-                                          Items_, Alloc_, Index_type_>;
+                                          Items_, Alloc_>;
     using Use_index=CGAL::Tag_true;
-    using Index_type=Index_type_;
-
-    typedef typename Traits_::Point  Point;
-    typedef typename Traits_::Vector Vector;
-    typedef typename Traits_::FT     FT;
+    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;
@@ -78,6 +76,8 @@ 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>
@@ -503,18 +503,18 @@ namespace CGAL {
 
   /// null_dart_handle
   template<unsigned int d_,  unsigned int ambient_dim, class Traits_,
-           class Items_, class Alloc_, class Index_type_>
-  typename CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_, Index_type_>::
-  Dart_index CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_, Index_type_>::
+           class Items_, class Alloc_>
+  typename CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
+  Dart_index CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
   null_dart_handle(0);
 
   /// null_handle
   template<unsigned int d_, unsigned int ambient_dim, class Traits_,
-           class Items_, class Alloc_, class Index_type_>
-  typename CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_, Index_type_>::
+           class Items_, class Alloc_>
+  typename CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
   Null_handle_type CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
-                                                      Items_, Alloc_, Index_type_>::
-  null_handle((std::numeric_limits<Index_type_>::max)()/2);
+                                                      Items_, Alloc_>::
+  null_handle((std::numeric_limits<Index_type>::max)()/2);
 
 } // namespace CGAL
 
diff --git a/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages.h b/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages.h
index 6368d7c9e30..40d3d5b184b 100644
--- a/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages.h
+++ b/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages.h
@@ -40,13 +40,14 @@ namespace CGAL {
   // as template parameter of Dart_wrapper. If we inherit, Self is not
   // the correct type).
   template<unsigned int d_, unsigned int ambient_dim,
-           class Traits_, class Items_, class Alloc_, class Concurrent_tag >
+           class Traits_, class Items_, class Alloc_>
   class GMap_linear_cell_complex_storage_1
   {
   public:
     using Self=GMap_linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
-    Items_, Alloc_, Concurrent_tag>;
+    Items_, Alloc_>;
     using Use_index=CGAL::Tag_false;
+    using Concurrent_tag=typename internal::Get_concurrent_tag<Items_>::type;
 
     typedef typename Traits_::Point  Point;
     typedef typename Traits_::Vector Vector;
@@ -75,6 +76,8 @@ 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>
@@ -467,11 +470,11 @@ namespace CGAL {
 
   /// null_handle
   template <unsigned int d_, unsigned int ambient_dim,
-           class Traits_, class Items_, class Alloc_, class Concurrent_tag >
+           class Traits_, class Items_, class Alloc_>
   const typename GMap_linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
-                                         Items_, Alloc_, Concurrent_tag>::Null_handle_type
+                                         Items_, Alloc_>::Null_handle_type
   GMap_linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
-                                Items_, Alloc_, Concurrent_tag>::null_handle = nullptr;
+                                Items_, Alloc_>::null_handle = nullptr;
 } // namespace CGAL
 
 #if defined(BOOST_GCC)
diff --git a/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages_with_index.h b/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages_with_index.h
index a90309c3fcd..febe11c38db 100644
--- a/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages_with_index.h
+++ b/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages_with_index.h
@@ -38,26 +38,28 @@ namespace CGAL {
   // as template parameter of Dart_wrapper. If we inherit, Self is not
   // the correct type).
   template<unsigned int d_, unsigned int ambient_dim, class Traits_,
-           class Items_, class Alloc_, class Index_type_ >
+           class Items_, class Alloc_>
   class GMap_linear_cell_complex_storage_2
   {
   public:
     using Self=GMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
-                                          Items_, Alloc_, Index_type_>;
+                                          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 Traits_::Point  Point;
     typedef typename Traits_::Vector Vector;
     typedef typename Traits_::FT     FT;
 
-    typedef internal::Combinatorial_map_helper<Self> Helper;
-
     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;
@@ -66,8 +68,6 @@ namespace CGAL {
     Multiply_by_two_policy_for_cc_with_size<64>, Index_type>
     Dart_container;
 
-    // typedef unsigned int Dart_index;
-    // typedef MyIndex<unsigned int> Dart_index;
     typedef typename Dart_container::Index Dart_index;
 
     // Definition of old types, for backward compatibility.
@@ -78,6 +78,8 @@ 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>
@@ -133,7 +135,7 @@ namespace CGAL {
     
     // Init
     void init_storage()
-    { null_dart_handle=nullptr; }
+    {}
 
     void clear_storage()
     {}
@@ -421,13 +423,13 @@ 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;
     }
 
@@ -438,12 +440,12 @@ 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;
     }
 
@@ -459,11 +461,11 @@ namespace CGAL {
 
   /// null_handle
   template<unsigned int d_, unsigned int ambient_dim, class Traits_,
-           class Items_, class Alloc_, class Index_type_>
-  typename GMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_, Index_type_>::
+           class Items_, class Alloc_>
+  typename GMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
   Null_handle_type GMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
-                                             Items_, Alloc_, Index_type_>::
-  null_handle((std::numeric_limits<Index_type_>::max)()/2);
+                                             Items_, Alloc_>::
+  null_handle((std::numeric_limits<Index_type>::max)()/2);
 
 } // namespace CGAL