diff --git a/Combinatorial_map/include/CGAL/Cell_attribute.h b/Combinatorial_map/include/CGAL/Cell_attribute.h
index 5bc4f045a45..02f2169e081 100644
--- a/Combinatorial_map/include/CGAL/Cell_attribute.h
+++ b/Combinatorial_map/include/CGAL/Cell_attribute.h
@@ -428,9 +428,6 @@ namespace CGAL {
     template < unsigned int, class, class, class, class >
     friend class Combinatorial_map_base;
 
-    template <unsigned int, typename>
-    friend struct Index::Dart;
-
     template < unsigned int, class, class, class, class >
     friend class Generalized_map_base;
 
@@ -439,23 +436,10 @@ namespace CGAL {
 
     template<unsigned int, unsigned int, class, class, class, class>
     friend class GMap_linear_cell_complex_storage_1;
->>>>>>> master
-
-    template < unsigned int, typename>
-    friend struct GMap_dart;
 
     template <class, class, class, class>
     friend class Compact_container_with_index;
 
-    template<typename, unsigned int, typename>
-    friend struct internal::Decrease_attribute_functor_run;
-
-    template <typename, typename>
-    friend struct internal::Reverse_orientation_of_map_functor;
-
-    template <typename, typename>
-    friend struct internal::Reverse_orientation_of_connected_component_functor;
-
   public:
     typedef Tag_false                            Supports_cell_dart;
 
@@ -544,27 +528,12 @@ namespace CGAL {
     template < unsigned int, class, class, class, class >
     friend class Combinatorial_map_base;
 
-    template <unsigned int, typename>
-    friend struct Index::Dart;
-
     template < unsigned int, class, class, class, class >
     friend class Generalized_map_base;
 
-    template <unsigned int, typename>
-    friend struct GMap_dart;
-
     template <class, class, class, class>
     friend class Compact_container_with_index;
 
-    template<typename, unsigned int, typename>
-    friend struct internal::Decrease_attribute_functor_run;
-
-    template <typename, typename>
-    friend struct internal::Reverse_orientation_of_map_functor;
-
-    template <typename, typename>
-    friend struct internal::Reverse_orientation_of_connected_component_functor;
-
   public:
     typedef Tag_true                             Supports_cell_dart;
 
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map/internal/Combinatorial_map_utility.h b/Combinatorial_map/include/CGAL/Combinatorial_map/internal/Combinatorial_map_utility.h
index db5d567551f..a460ecd9867 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map/internal/Combinatorial_map_utility.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map/internal/Combinatorial_map_utility.h
@@ -435,9 +435,10 @@ namespace CGAL
         iterator_type;
 
         // TODO case when there is no Use_index typedef in CMap
-        typedef typename boost::mpl::if_
+        /* TODO ask sebastien ?
+         typedef typename boost::mpl::if_
         < typename boost::is_same<typename CMap::Use_index,Tag_true>::type,
-          typename CMap::Dart_handle, iterator_type >::type type;
+          typename CMap::Dart_handle, iterator_type >::type type;*/
 
         typedef typename GetIndexOrHandle
         <typename CMap::template Container_for_attributes<T> >::type type;
@@ -451,10 +452,11 @@ namespace CGAL
         typedef typename CMap::template Container_for_attributes<T>::
         const_iterator iterator_type;
 
+        /* TODO ask sebastien ?
         typedef typename boost::mpl::if_
              < typename boost::is_same<typename CMap::Use_index,Tag_true>::type,
                typename CMap::Dart_handle, iterator_type >::type type;
-
+         */
         typedef typename GetConstIndexOrConstHandle
         <typename CMap::template Container_for_attributes<T> >::type type;
       };
diff --git a/Combinatorial_map/include/CGAL/Dart.h b/Combinatorial_map/include/CGAL/Dart.h
index 4855f70795e..0ff820763d3 100644
--- a/Combinatorial_map/include/CGAL/Dart.h
+++ b/Combinatorial_map/include/CGAL/Dart.h
@@ -29,12 +29,18 @@ namespace CGAL {
   template<unsigned int, class, class, class>
   class Combinatorial_map_storage_1;
 
+  template<unsigned int, class, class, class>
+  class Combinatorial_map_storage_2;
+
   template<unsigned int, class, class, class>
   class Generalized_map_storage_1;
 
   template<unsigned int, unsigned int, class, class, class, class>
   class CMap_linear_cell_complex_storage_1;
 
+  template<unsigned int, unsigned int, class, class, class, class>
+  class CMap_linear_cell_complex_storage_2;
+
   template<unsigned int, unsigned int, class, class, class, class>
   class GMap_linear_cell_complex_storage_1;
 
@@ -264,14 +270,12 @@ namespace CGAL {
   public:
     typedef CGAL::Void Info;
   };
+
   namespace Index
   {
   template <unsigned int d, typename Refs>
   struct Dart
   {
-    template < unsigned int, class, class, class, class >
-    friend class Combinatorial_map_base;
-
     template<unsigned int, class, class>
     friend class Combinatorial_map_storage_1;
 
@@ -287,27 +291,15 @@ namespace CGAL {
     template <class, class, class, class>
     friend class Compact_container;
 
+    template <class, class>
+    friend class Concurrent_compact_container;
+
     template <class, class, class, class>
     friend class Compact_container_with_index;
 
     template <class, class, class, class>
     friend class Compact_container_with_index_2;
 
-    template<class, unsigned int, unsigned int>
-    friend struct Remove_cell_functor;
-
-    template<class, unsigned int>
-    friend struct Contract_cell_functor;
-
-    template <typename,unsigned int>
-    friend struct internal::link_beta_functor;
-
-    template <typename, typename>
-    friend struct internal::Reverse_orientation_of_map_functor;
-
-    template <typename, typename>
-    friend struct internal::Reverse_orientation_of_connected_component_functor;
-
   public:
     typedef Dart<d,Refs>                     Self;
     typedef typename Refs::Dart_handle       Dart_handle;
@@ -329,47 +321,11 @@ namespace CGAL {
     /// The dimension of the combinatorial map.
     static const unsigned int dimension = d;
 
-    /** Return the beta of this dart for a given dimension.
-     * @param i the dimension.
-     * @return beta(\em i).
-     */
-    template<unsigned int i>
-    Dart_handle beta()
+    Dart_handle get_f(unsigned int i) const
     {
-      CGAL_assertion(i <= dimension);
-      return mbeta[i];
+      assert(i<=dimension);
+      return mf[i];
     }
-    Dart_handle beta(unsigned int i)
-    {
-      CGAL_assertion(i <= dimension);
-      return mbeta[i];
-    }
-    template<unsigned int i>
-    Dart_const_handle beta() const
-    {
-      CGAL_assertion(i <= dimension);
-      return mbeta[i];
-    }
-    Dart_const_handle beta(unsigned int i) const
-    {
-      CGAL_assertion(i <= dimension);
-      return mbeta[i];
-    }
-
-    /** Return the beta inverse of this dart for a given dimension.
-     * @param i the dimension.
-     * @return beta^{-1}(\em i).
-     */
-    template<unsigned int i>
-    Dart_handle beta_inv()
-    { return beta<CGAL_BETAINV(i)>(); }
-    Dart_handle beta_inv(unsigned int i)
-    { return beta(CGAL_BETAINV(i)); }
-    template<unsigned int i>
-    Dart_const_handle beta_inv() const
-    { return beta<CGAL_BETAINV(i)>(); }
-    Dart_const_handle beta_inv(unsigned int i) const
-    { return beta(CGAL_BETAINV(i)); }
 
     /// @return a handle on the i-attribute
     template<int i>
@@ -455,18 +411,18 @@ namespace CGAL {
       mattribute_handles(adart.mattribute_handles)
     {
       for (unsigned int i = 0; i <= dimension; ++i)
-        mbeta[i] = adart.mbeta[i];
+        mf[i] = adart.mf[i];
     }
 
   public:
     size_type for_compact_container_with_index() const
-    { return mbeta[0].for_compact_container_with_index(); }
+    { return mf[0].for_compact_container_with_index(); }
     size_type& for_compact_container_with_index()
-    { return mbeta[0].for_compact_container_with_index(); }
+    { return mf[0].for_compact_container_with_index(); }
 
   protected:
-    /// Beta for each dimension +1 (from 0 to dimension).
-    Dart_handle mbeta[dimension+1];
+    /// Neighboors for each dimension +1 (from 0 to dimension).
+    Dart_handle mf[dimension+1];
 
     /// Values of Boolean marks.
     mutable std::bitset<NB_MARKS> mmarks;
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex_fwd.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex_fwd.h
index 9ee33103833..b130f582a65 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex_fwd.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex_fwd.h
@@ -26,6 +26,14 @@ template<unsigned int d_, unsigned int ambient_dim,
          class Traits_, class Items_, class Alloc_, class Concurrent_tag=CGAL::Tag_false >
 class GMap_linear_cell_complex_storage_1;
 
+template<unsigned int d_, unsigned int ambient_dim,
+         class Traits_, class Items_, class Alloc_, class Size_type> // TODO add Concurrent_tag
+class CMap_linear_cell_complex_storage_2;
+
+template<unsigned int d_, unsigned int ambient_dim,
+         class Traits_, class Items_, class Alloc_, class Size_type> // TODO add Concurrent_tag class Concurrent_tag=CGAL::Tag_false >
+class GMap_linear_cell_complex_storage_2;
+
 template <unsigned int d>
 struct LCC_default_kernel;
 
@@ -67,6 +75,33 @@ template < unsigned int d_, unsigned int ambient_dim = d_,
                                                                Alloc_, CGAL::Tag_false> >
 class Linear_cell_complex_for_generalized_map;
 
+    namespace Index
+    {
+    template < unsigned int d_, unsigned int ambient_dim = d_,
+               class Traits_ = Linear_cell_complex_traits<ambient_dim>,
+               class Items_ = Linear_cell_complex_min_items,
+               class Alloc_ = CGAL_ALLOCATOR(int),
+               template<unsigned int,class,class,class,class>
+               class CMap = Combinatorial_map_base,
+               class Storage_ = CMap_linear_cell_complex_storage_2<d_, ambient_dim,
+                                                                   Traits_, Items_,
+                                                                   Alloc_, unsigned int>>
+                                                                   // TODO READD CGAL::Tag_false> >
+      class Linear_cell_complex_for_combinatorial_map;
+
+      /* TODO GMap_linear_cell_complex_storage_2
+       template < unsigned int d_, unsigned int ambient_dim = d_,
+                 class Traits_ = Linear_cell_complex_traits<ambient_dim>,
+                 class Items_ = Linear_cell_complex_min_items,
+                 class Alloc_ = CGAL_ALLOCATOR(int),
+                 template<unsigned int,class,class,class,class>
+                 class CMap = Generalized_map_base,
+                 class Storage_ = GMap_linear_cell_complex_storage_2<d_, ambient_dim,
+                                                                     Traits_, Items_, unsigned int>>
+      // TODO READD CGAL::Tag_false> >
+        class Linear_cell_complex_for_generalized_map; */
+    } // namespace Index
+
 } // CGAL
 
 #endif // LINEAR_CELL_COMPLEX_FWD_H
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex_min_items.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex_min_items.h
index 37320ac326a..247556e37f6 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex_min_items.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex_min_items.h
@@ -40,7 +40,6 @@ namespace CGAL {
 
   namespace Index
   {
-  template <unsigned int d>
   struct Linear_cell_complex_min_items
   {
     /// Dart_wrapper defines the type of darts used.