diff --git a/vtkm/cont/ArrayHandleGroupVecVariable.h b/vtkm/cont/ArrayHandleGroupVecVariable.h
index 5e2f78997..d7b19347d 100644
--- a/vtkm/cont/ArrayHandleGroupVecVariable.h
+++ b/vtkm/cont/ArrayHandleGroupVecVariable.h
@@ -119,10 +119,9 @@ namespace arg
 // Instead, you need to implement the Load to point to the array portal. You
 // can also ignore the Store because the data is already set in the array at
 // that point.
-template <typename ThreadIndicesType, typename SourcePortalType, typename OffsetsPortalType>
+template <typename SourcePortalType, typename OffsetsPortalType>
 struct Fetch<vtkm::exec::arg::FetchTagArrayDirectOut,
              vtkm::exec::arg::AspectTagDefault,
-             ThreadIndicesType,
              vtkm::exec::internal::ArrayPortalGroupVecVariable<SourcePortalType, OffsetsPortalType>>
 {
   using ExecObjectType =
@@ -130,15 +129,16 @@ struct Fetch<vtkm::exec::arg::FetchTagArrayDirectOut,
   using ValueType = typename ExecObjectType::ValueType;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType& arrayPortal) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices,
+                           const ExecObjectType& arrayPortal) const
   {
     return arrayPortal.Get(indices.GetOutputIndex());
   }
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // We can actually ignore this because the VecFromPortal will already have
     // set new values in the array.
diff --git a/vtkm/exec/arg/Boundary.h b/vtkm/exec/arg/Boundary.h
index df3612a3e..526837874 100644
--- a/vtkm/exec/arg/Boundary.h
+++ b/vtkm/exec/arg/Boundary.h
@@ -40,24 +40,20 @@ struct Boundary : vtkm::exec::arg::ExecutionSignatureTagBase
 };
 
 template <typename FetchTag, typename ExecObjectType>
-struct Fetch<FetchTag,
-             vtkm::exec::arg::AspectTagBoundary,
-             vtkm::exec::arg::ThreadIndicesPointNeighborhood,
-             ExecObjectType>
+struct Fetch<FetchTag, vtkm::exec::arg::AspectTagBoundary, ExecObjectType>
 {
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesPointNeighborhood;
 
   using ValueType = vtkm::exec::BoundaryState;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices, const ExecObjectType&) const
   {
     return indices.GetBoundaryState();
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op.
   }
diff --git a/vtkm/exec/arg/CellShape.h b/vtkm/exec/arg/CellShape.h
index a5660283d..83d813315 100644
--- a/vtkm/exec/arg/CellShape.h
+++ b/vtkm/exec/arg/CellShape.h
@@ -12,7 +12,6 @@
 
 #include <vtkm/exec/arg/ExecutionSignatureTagBase.h>
 #include <vtkm/exec/arg/Fetch.h>
-#include <vtkm/exec/arg/ThreadIndicesTopologyMap.h>
 
 namespace vtkm
 {
@@ -38,25 +37,19 @@ struct CellShape : vtkm::exec::arg::ExecutionSignatureTagBase
   using AspectTag = vtkm::exec::arg::AspectTagCellShape;
 };
 
-template <typename FetchTag, typename ConnectivityType, typename ExecObjectType>
-struct Fetch<FetchTag,
-             vtkm::exec::arg::AspectTagCellShape,
-             vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>,
-             ExecObjectType>
+template <typename FetchTag, typename ExecObjectType>
+struct Fetch<FetchTag, vtkm::exec::arg::AspectTagCellShape, ExecObjectType>
 {
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>;
-
-  using ValueType = typename ThreadIndicesType::CellShapeTag;
-
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC auto Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+    -> decltype(indices.GetCellShape())
   {
     return indices.GetCellShape();
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType, typename ValueType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op.
   }
diff --git a/vtkm/exec/arg/Fetch.h b/vtkm/exec/arg/Fetch.h
index 9fbfcde3e..5d23ec4d5 100644
--- a/vtkm/exec/arg/Fetch.h
+++ b/vtkm/exec/arg/Fetch.h
@@ -45,10 +45,7 @@ namespace arg
 /// ExecutionSignature with a particular aspect is pointing to the wrong
 /// argument or an invalid argument in the ControlSignature.
 ///
-template <typename FetchTag,
-          typename AspectTag,
-          typename ThreadIndicesType,
-          typename ExecObjectType>
+template <typename FetchTag, typename AspectTag, typename ExecObjectType>
 struct Fetch
 #ifdef VTKM_DOXYGEN_ONLY
 {
@@ -68,8 +65,9 @@ struct Fetch
   /// work instance. If there is no actual data to load (for example for a
   /// write-only fetch), this method can be a no-op and return any value.
   ///
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType& execObject) const;
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices,
+                           const ExecObjectType& execObject) const;
 
   /// \brief Store data from a work instance.
   ///
@@ -80,10 +78,10 @@ struct Fetch
   /// fetch. If the store is not applicable (for example for a read-only
   /// fetch), this method can be a no-op.
   ///
-  VTKM_EXEC
-  void Store(const ThreadIndicesType& indices,
-             const ExecObjectType& execObject,
-             const ValueType& value) const;
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType& indices,
+                       const ExecObjectType& execObject,
+                       const ValueType& value) const;
 };
 #else  // VTKM_DOXYGEN_ONLY
   ;
diff --git a/vtkm/exec/arg/FetchExtrude.h b/vtkm/exec/arg/FetchExtrude.h
index ecc6cd9fa..05bac2a33 100644
--- a/vtkm/exec/arg/FetchExtrude.h
+++ b/vtkm/exec/arg/FetchExtrude.h
@@ -26,26 +26,20 @@ namespace arg
 {
 
 //Optimized fetch for point ids when iterating the cells ConnectivityExtrude
-template <typename FetchType, typename Device, typename ExecObjectType>
-struct Fetch<FetchType,
-             vtkm::exec::arg::AspectTagIncidentElementIndices,
-             vtkm::exec::arg::ThreadIndicesTopologyMap<vtkm::exec::ConnectivityExtrude<Device>>,
-             ExecObjectType>
+template <typename FetchType, typename ExecObjectType>
+struct Fetch<FetchType, vtkm::exec::arg::AspectTagIncidentElementIndices, ExecObjectType>
 {
-  using ThreadIndicesType =
-    vtkm::exec::arg::ThreadIndicesTopologyMap<vtkm::exec::ConnectivityExtrude<Device>>;
-
-  using ValueType = vtkm::Vec<vtkm::Id, 6>;
-
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename Device>
+  VTKM_EXEC auto Load(const vtkm::exec::arg::ThreadIndicesTopologyMap<
+                        vtkm::exec::ConnectivityExtrude<Device>>& indices,
+                      const ExecObjectType&) const -> vtkm::Vec<vtkm::Id, 6>
   {
     // std::cout << "opimized fetch for point ids" << std::endl;
     const auto& xgcidx = indices.GetIndicesIncident();
     const vtkm::Id offset1 = (xgcidx.Planes[0] * xgcidx.NumberOfPointsPerPlane);
     const vtkm::Id offset2 = (xgcidx.Planes[1] * xgcidx.NumberOfPointsPerPlane);
-    ValueType result;
+    vtkm::Vec<vtkm::Id, 6> result;
     result[0] = offset1 + xgcidx.PointIds[0][0];
     result[1] = offset1 + xgcidx.PointIds[0][1];
     result[2] = offset1 + xgcidx.PointIds[0][2];
@@ -55,132 +49,104 @@ struct Fetch<FetchType,
     return result;
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  VTKM_SUPPRESS_EXEC_WARNINGS
+  template <typename ConnectivityType>
+  VTKM_EXEC auto Load(const vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>& indices,
+                      const ExecObjectType&) const -> decltype(indices.GetIndicesIncident())
+  {
+    return indices.GetIndicesIncident();
+  }
+
+  template <typename ThreadIndicesType, typename ValueType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op.
   }
 };
 
-//Optimized fetch for point arrays when iterating the cells ConnectivityExtrude
-template <typename Device, typename PortalType>
-struct Fetch<vtkm::exec::arg::FetchTagArrayTopologyMapIn,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesTopologyMap<vtkm::exec::ConnectivityExtrude<Device>>,
-             PortalType>
-{
-  using ThreadIndicesType =
-    vtkm::exec::arg::ThreadIndicesTopologyMap<vtkm::exec::ConnectivityExtrude<Device>>;
-  using ValueType = vtkm::Vec<typename PortalType::ValueType, 6>;
-
-  VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const PortalType& portal)
-  {
-    // std::cout << "opimized fetch for point values" << std::endl;
-    const auto& xgcidx = indices.GetIndicesIncident();
-    const vtkm::Id offset1 = (xgcidx.Planes[0] * xgcidx.NumberOfPointsPerPlane);
-    const vtkm::Id offset2 = (xgcidx.Planes[1] * xgcidx.NumberOfPointsPerPlane);
-    ValueType result;
-    result[0] = portal.Get(offset1 + xgcidx.PointIds[0][0]);
-    result[1] = portal.Get(offset1 + xgcidx.PointIds[0][1]);
-    result[2] = portal.Get(offset1 + xgcidx.PointIds[0][2]);
-    result[3] = portal.Get(offset2 + xgcidx.PointIds[1][0]);
-    result[4] = portal.Get(offset2 + xgcidx.PointIds[1][1]);
-    result[5] = portal.Get(offset2 + xgcidx.PointIds[1][2]);
-    return result;
-  }
-
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const PortalType&, const ValueType&) const
-  {
-    // Store is a no-op for this fetch.
-  }
-};
-
 //Optimized fetch for point coordinates when iterating the cells of ConnectivityExtrude
-template <typename Device, typename T>
+template <typename T>
 struct Fetch<vtkm::exec::arg::FetchTagArrayTopologyMapIn,
              vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesTopologyMap<vtkm::exec::ConnectivityExtrude<Device>>,
              vtkm::exec::ArrayPortalExtrude<T>>
 
 {
-  using ThreadIndicesType =
-    vtkm::exec::arg::ThreadIndicesTopologyMap<vtkm::exec::ConnectivityExtrude<Device>>;
-  using ValueType = vtkm::Vec<typename vtkm::exec::ArrayPortalExtrude<T>::ValueType, 6>;
-
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const vtkm::exec::ArrayPortalExtrude<T>& points)
+  template <typename ThreadIndicesType>
+  VTKM_EXEC auto Load(const ThreadIndicesType& indices,
+                      const vtkm::exec::ArrayPortalExtrude<T>& points)
+    -> decltype(points.GetWedge(indices.GetIndicesIncident()))
   {
     // std::cout << "opimized fetch for point coordinates" << std::endl;
     return points.GetWedge(indices.GetIndicesIncident());
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&,
-             const vtkm::exec::ArrayPortalExtrude<T>&,
-             const ValueType&) const
+  template <typename ThreadIndicesType, typename ValueType>
+  VTKM_EXEC void Store(const ThreadIndicesType&,
+                       const vtkm::exec::ArrayPortalExtrude<T>&,
+                       const ValueType&) const
   {
     // Store is a no-op for this fetch.
   }
 };
 
 //Optimized fetch for point coordinates when iterating the cells of ConnectivityExtrude
-template <typename Device, typename T>
+template <typename T>
 struct Fetch<vtkm::exec::arg::FetchTagArrayTopologyMapIn,
              vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesTopologyMap<vtkm::exec::ConnectivityExtrude<Device>>,
              vtkm::exec::ArrayPortalExtrudePlane<T>>
 {
-  using ThreadIndicesType =
-    vtkm::exec::arg::ThreadIndicesTopologyMap<vtkm::exec::ConnectivityExtrude<Device>>;
-  using ValueType = vtkm::Vec<typename vtkm::exec::ArrayPortalExtrudePlane<T>::ValueType, 6>;
-
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices,
-                 const vtkm::exec::ArrayPortalExtrudePlane<T>& portal)
+  template <typename ThreadIndicesType>
+  VTKM_EXEC auto Load(const ThreadIndicesType& indices,
+                      const vtkm::exec::ArrayPortalExtrudePlane<T>& portal)
+    -> decltype(portal.GetWedge(indices.GetIndicesIncident()))
   {
     // std::cout << "opimized fetch for point coordinates" << std::endl;
     return portal.GetWedge(indices.GetIndicesIncident());
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&,
-             const vtkm::exec::ArrayPortalExtrudePlane<T>&,
-             const ValueType&) const
+  template <typename ThreadIndicesType, typename ValueType>
+  VTKM_EXEC void Store(const ThreadIndicesType&,
+                       const vtkm::exec::ArrayPortalExtrudePlane<T>&,
+                       const ValueType&) const
   {
     // Store is a no-op for this fetch.
   }
 };
 
 //Optimized fetch for point coordinates when iterating the points of ConnectivityExtrude
-template <typename Device, typename T>
-struct Fetch<
-  vtkm::exec::arg::FetchTagArrayDirectIn,
-  vtkm::exec::arg::AspectTagDefault,
-  vtkm::exec::arg::ThreadIndicesTopologyMap<vtkm::exec::ReverseConnectivityExtrude<Device>>,
-  vtkm::exec::ArrayPortalExtrude<T>>
+template <typename T>
+struct Fetch<vtkm::exec::arg::FetchTagArrayDirectIn,
+             vtkm::exec::arg::AspectTagDefault,
+             vtkm::exec::ArrayPortalExtrude<T>>
 
 {
-  using ThreadIndicesType =
-    vtkm::exec::arg::ThreadIndicesTopologyMap<vtkm::exec::ReverseConnectivityExtrude<Device>>;
-  using ValueType = typename vtkm::exec::ArrayPortalExtrude<T>::ValueType;
+  VTKM_SUPPRESS_EXEC_WARNINGS
+  template <typename ThreadIndicesType>
+  VTKM_EXEC auto Load(const ThreadIndicesType& indices,
+                      const vtkm::exec::ArrayPortalExtrude<T>& points)
+    -> decltype(points.Get(indices.GetInputIndex()))
+  {
+    // std::cout << "optimized fetch for point coordinates" << std::endl;
+    return points.Get(indices.GetInputIndex());
+  }
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const vtkm::exec::ArrayPortalExtrude<T>& points)
+  template <typename Device>
+  VTKM_EXEC auto Load(const vtkm::exec::arg::ThreadIndicesTopologyMap<
+                        vtkm::exec::ReverseConnectivityExtrude<Device>>& indices,
+                      const vtkm::exec::ArrayPortalExtrude<T>& points)
+    -> decltype(points.Get(indices.GetIndexLogical()))
   {
-    // std::cout << "opimized fetch for point coordinates" << std::endl;
+    // std::cout << "optimized fetch for point coordinates" << std::endl;
     return points.Get(indices.GetIndexLogical());
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&,
-             const vtkm::exec::ArrayPortalExtrude<T>&,
-             const ValueType&) const
+  template <typename ThreadIndicesType, typename ValueType>
+  VTKM_EXEC void Store(const ThreadIndicesType&,
+                       const vtkm::exec::ArrayPortalExtrude<T>&,
+                       const ValueType&) const
   {
     // Store is a no-op for this fetch.
   }
diff --git a/vtkm/exec/arg/FetchTagArrayDirectIn.h b/vtkm/exec/arg/FetchTagArrayDirectIn.h
index 851e8d899..959220a8c 100644
--- a/vtkm/exec/arg/FetchTagArrayDirectIn.h
+++ b/vtkm/exec/arg/FetchTagArrayDirectIn.h
@@ -45,10 +45,9 @@ inline VTKM_EXEC T load(const U* u, vtkm::Id v)
   return u->Get(v);
 }
 
-template <typename ThreadIndicesType, typename ExecObjectType>
+template <typename ExecObjectType>
 struct Fetch<vtkm::exec::arg::FetchTagArrayDirectIn,
              vtkm::exec::arg::AspectTagDefault,
-             ThreadIndicesType,
              ExecObjectType>
 {
   //need to remove pointer type from ThreadIdicesType
@@ -59,14 +58,14 @@ struct Fetch<vtkm::exec::arg::FetchTagArrayDirectIn,
   using ValueType = typename ET::ValueType;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, PortalType arrayPortal) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices, PortalType arrayPortal) const
   {
     return load<ValueType>(arrayPortal, indices.GetInputIndex());
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, PortalType, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, PortalType, const ValueType&) const
   {
     // Store is a no-op for this fetch.
   }
diff --git a/vtkm/exec/arg/FetchTagArrayDirectInOut.h b/vtkm/exec/arg/FetchTagArrayDirectInOut.h
index b6129007b..f6f7abad6 100644
--- a/vtkm/exec/arg/FetchTagArrayDirectInOut.h
+++ b/vtkm/exec/arg/FetchTagArrayDirectInOut.h
@@ -40,26 +40,26 @@ struct FetchTagArrayDirectInOut
 {
 };
 
-template <typename ThreadIndicesType, typename ExecObjectType>
+template <typename ExecObjectType>
 struct Fetch<vtkm::exec::arg::FetchTagArrayDirectInOut,
              vtkm::exec::arg::AspectTagDefault,
-             ThreadIndicesType,
              ExecObjectType>
 {
   using ValueType = typename ExecObjectType::ValueType;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType& arrayPortal) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices,
+                           const ExecObjectType& arrayPortal) const
   {
     return arrayPortal.Get(indices.GetOutputIndex());
   }
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  void Store(const ThreadIndicesType& indices,
-             const ExecObjectType& arrayPortal,
-             const ValueType& value) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType& indices,
+                       const ExecObjectType& arrayPortal,
+                       const ValueType& value) const
   {
     arrayPortal.Set(indices.GetOutputIndex(), value);
   }
diff --git a/vtkm/exec/arg/FetchTagArrayDirectOut.h b/vtkm/exec/arg/FetchTagArrayDirectOut.h
index 1a429b61a..1ca3d997e 100644
--- a/vtkm/exec/arg/FetchTagArrayDirectOut.h
+++ b/vtkm/exec/arg/FetchTagArrayDirectOut.h
@@ -30,16 +30,15 @@ struct FetchTagArrayDirectOut
 {
 };
 
-template <typename ThreadIndicesType, typename ExecObjectType>
+template <typename ExecObjectType>
 struct Fetch<vtkm::exec::arg::FetchTagArrayDirectOut,
              vtkm::exec::arg::AspectTagDefault,
-             ThreadIndicesType,
              ExecObjectType>
 {
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  auto Load(const ThreadIndicesType&, const ExecObjectType&) ->
-    typename ExecObjectType::ValueType const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC auto Load(const ThreadIndicesType&, const ExecObjectType&) const ->
+    typename ExecObjectType::ValueType
   {
     // Load is a no-op for this fetch.
     using ValueType = typename ExecObjectType::ValueType;
@@ -47,7 +46,7 @@ struct Fetch<vtkm::exec::arg::FetchTagArrayDirectOut,
   }
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  template <typename T>
+  template <typename ThreadIndicesType, typename T>
   VTKM_EXEC void Store(const ThreadIndicesType& indices,
                        const ExecObjectType& arrayPortal,
                        const T& value) const
diff --git a/vtkm/exec/arg/FetchTagArrayNeighborhoodIn.h b/vtkm/exec/arg/FetchTagArrayNeighborhoodIn.h
index 86ee7f028..6964b55b0 100644
--- a/vtkm/exec/arg/FetchTagArrayNeighborhoodIn.h
+++ b/vtkm/exec/arg/FetchTagArrayNeighborhoodIn.h
@@ -34,21 +34,20 @@ struct FetchTagArrayNeighborhoodIn
 template <typename ExecObjectType>
 struct Fetch<vtkm::exec::arg::FetchTagArrayNeighborhoodIn,
              vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesPointNeighborhood,
              ExecObjectType>
 {
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesPointNeighborhood;
   using ValueType = vtkm::exec::FieldNeighborhood<ExecObjectType>;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType& arrayPortal) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices,
+                           const ExecObjectType& arrayPortal) const
   {
     return ValueType(arrayPortal, indices.GetBoundaryState());
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op for this fetch.
   }
diff --git a/vtkm/exec/arg/FetchTagArrayTopologyMapIn.h b/vtkm/exec/arg/FetchTagArrayTopologyMapIn.h
index b99664af2..28af1c867 100644
--- a/vtkm/exec/arg/FetchTagArrayTopologyMapIn.h
+++ b/vtkm/exec/arg/FetchTagArrayTopologyMapIn.h
@@ -19,6 +19,7 @@
 #include <vtkm/internal/ArrayPortalUniformPointCoordinates.h>
 
 #include <vtkm/VecAxisAlignedPointCoordinates.h>
+#include <vtkm/exec/ConnectivityExtrude.h>
 #include <vtkm/exec/ConnectivityStructured.h>
 
 #include <vtkm/VecFromPortalPermute.h>
@@ -51,12 +52,9 @@ namespace detail
 // and the field is regular point coordinates, it is much faster to compute the
 // field directly.
 
-template <typename ConnectivityType, typename FieldExecObjectType>
+template <typename ConnectivityType, typename FieldExecObjectType, typename ThreadIndicesType>
 struct FetchArrayTopologyMapInImplementation
 {
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>;
-
-  // ThreadIndicesTopologyMap has special incident element indices that are
   // stored in a Vec-like object.
   using IndexVecType = typename ThreadIndicesType::IndicesIncidentType;
 
@@ -128,18 +126,18 @@ static inline VTKM_EXEC vtkm::VecAxisAlignedPointCoordinates<3> make_VecAxisAlig
   return vtkm::VecAxisAlignedPointCoordinates<3>(offsetOrigin, spacing);
 }
 
-template <vtkm::IdComponent NumDimensions>
+template <vtkm::IdComponent NumDimensions, typename ThreadIndicesType>
 struct FetchArrayTopologyMapInImplementation<
   vtkm::exec::ConnectivityStructured<vtkm::TopologyElementTagCell,
                                      vtkm::TopologyElementTagPoint,
                                      NumDimensions>,
-  vtkm::internal::ArrayPortalUniformPointCoordinates>
+  vtkm::internal::ArrayPortalUniformPointCoordinates,
+  ThreadIndicesType>
 
 {
   using ConnectivityType = vtkm::exec::ConnectivityStructured<vtkm::TopologyElementTagCell,
                                                               vtkm::TopologyElementTagPoint,
                                                               NumDimensions>;
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>;
 
   using ValueType = vtkm::VecAxisAlignedPointCoordinates<NumDimensions>;
 
@@ -155,14 +153,15 @@ struct FetchArrayTopologyMapInImplementation<
   }
 };
 
-template <typename PermutationPortal, vtkm::IdComponent NumDimensions>
+template <typename PermutationPortal, vtkm::IdComponent NumDimensions, typename ThreadIndicesType>
 struct FetchArrayTopologyMapInImplementation<
   vtkm::exec::ConnectivityPermutedVisitCellsWithPoints<
     PermutationPortal,
     vtkm::exec::ConnectivityStructured<vtkm::TopologyElementTagCell,
                                        vtkm::TopologyElementTagPoint,
                                        NumDimensions>>,
-  vtkm::internal::ArrayPortalUniformPointCoordinates>
+  vtkm::internal::ArrayPortalUniformPointCoordinates,
+  ThreadIndicesType>
 
 {
   using ConnectivityType = vtkm::exec::ConnectivityPermutedVisitCellsWithPoints<
@@ -170,7 +169,6 @@ struct FetchArrayTopologyMapInImplementation<
     vtkm::exec::ConnectivityStructured<vtkm::TopologyElementTagCell,
                                        vtkm::TopologyElementTagPoint,
                                        NumDimensions>>;
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>;
 
   using ValueType = vtkm::VecAxisAlignedPointCoordinates<NumDimensions>;
 
@@ -191,28 +189,56 @@ struct FetchArrayTopologyMapInImplementation<
 
 } // namespace detail
 
-template <typename ConnectivityType, typename ExecObjectType>
+template <typename ExecObjectType>
 struct Fetch<vtkm::exec::arg::FetchTagArrayTopologyMapIn,
              vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>,
              ExecObjectType>
 {
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>;
-
-  using Implementation =
-    detail::FetchArrayTopologyMapInImplementation<ConnectivityType, ExecObjectType>;
-
-  using ValueType = typename Implementation::ValueType;
 
+  //using ConnectivityType = typename ThreadIndicesType::Connectivity;
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType& field) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC auto Load(const ThreadIndicesType& indices, const ExecObjectType& field) const
+    -> decltype(
+      detail::FetchArrayTopologyMapInImplementation<typename ThreadIndicesType::Connectivity,
+                                                    ExecObjectType,
+                                                    ThreadIndicesType>::Load(indices, field))
   {
+    using Implementation =
+      detail::FetchArrayTopologyMapInImplementation<typename ThreadIndicesType::Connectivity,
+                                                    ExecObjectType,
+                                                    ThreadIndicesType>;
     return Implementation::Load(indices, field);
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  //Optimized fetch for point arrays when iterating the cells ConnectivityExtrude
+  VTKM_SUPPRESS_EXEC_WARNINGS
+  template <typename Device>
+  VTKM_EXEC auto Load(const vtkm::exec::arg::ThreadIndicesTopologyMap<
+                        vtkm::exec::ConnectivityExtrude<Device>>& indices,
+                      const ExecObjectType& portal)
+    -> vtkm::Vec<typename ExecObjectType::ValueType, 6>
+  {
+    // std::cout << "opimized fetch for point values" << std::endl;
+    const auto& xgcidx = indices.GetIndicesIncident();
+    const vtkm::Id offset1 = (xgcidx.Planes[0] * xgcidx.NumberOfPointsPerPlane);
+    const vtkm::Id offset2 = (xgcidx.Planes[1] * xgcidx.NumberOfPointsPerPlane);
+
+    using ValueType = vtkm::Vec<typename ExecObjectType::ValueType, 6>;
+    ValueType result;
+
+    result[0] = portal.Get(offset1 + xgcidx.PointIds[0][0]);
+    result[1] = portal.Get(offset1 + xgcidx.PointIds[0][1]);
+    result[2] = portal.Get(offset1 + xgcidx.PointIds[0][2]);
+    result[3] = portal.Get(offset2 + xgcidx.PointIds[1][0]);
+    result[4] = portal.Get(offset2 + xgcidx.PointIds[1][1]);
+    result[5] = portal.Get(offset2 + xgcidx.PointIds[1][2]);
+    return result;
+  }
+
+
+  template <typename ThreadIndicesType, typename T>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const T&) const
   {
     // Store is a no-op for this fetch.
   }
diff --git a/vtkm/exec/arg/FetchTagCellSetIn.h b/vtkm/exec/arg/FetchTagCellSetIn.h
index 05b1a3af5..cc0fedc29 100644
--- a/vtkm/exec/arg/FetchTagCellSetIn.h
+++ b/vtkm/exec/arg/FetchTagCellSetIn.h
@@ -12,7 +12,6 @@
 
 #include <vtkm/exec/arg/AspectTagDefault.h>
 #include <vtkm/exec/arg/Fetch.h>
-#include <vtkm/exec/arg/ThreadIndicesTopologyMap.h>
 
 namespace vtkm
 {
@@ -31,25 +30,19 @@ struct FetchTagCellSetIn
 {
 };
 
-template <typename ConnectivityType, typename ExecObjectType>
-struct Fetch<vtkm::exec::arg::FetchTagCellSetIn,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>,
-             ExecObjectType>
+template <typename ExecObjectType>
+struct Fetch<vtkm::exec::arg::FetchTagCellSetIn, vtkm::exec::arg::AspectTagDefault, ExecObjectType>
 {
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>;
-
-  using ValueType = typename ThreadIndicesType::CellShapeTag;
-
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC auto Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+    -> decltype(indices.GetCellShape())
   {
     return indices.GetCellShape();
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType, typename ValueType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op for this fetch.
   }
diff --git a/vtkm/exec/arg/FetchTagExecObject.h b/vtkm/exec/arg/FetchTagExecObject.h
index 6945354c8..f98c9cfe9 100644
--- a/vtkm/exec/arg/FetchTagExecObject.h
+++ b/vtkm/exec/arg/FetchTagExecObject.h
@@ -33,24 +33,21 @@ struct FetchTagExecObject
 {
 };
 
-template <typename ThreadIndicesType, typename ExecObjectType>
-struct Fetch<vtkm::exec::arg::FetchTagExecObject,
-             vtkm::exec::arg::AspectTagDefault,
-             ThreadIndicesType,
-             ExecObjectType>
+template <typename ExecObjectType>
+struct Fetch<vtkm::exec::arg::FetchTagExecObject, vtkm::exec::arg::AspectTagDefault, ExecObjectType>
 {
   using ValueType = ExecObjectType;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& vtkmNotUsed(indices),
-                 const ExecObjectType& execObject) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& vtkmNotUsed(indices),
+                           const ExecObjectType& execObject) const
   {
     return execObject;
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op for this fetch.
   }
diff --git a/vtkm/exec/arg/FetchTagKeysIn.h b/vtkm/exec/arg/FetchTagKeysIn.h
index 0f1076391..e3a540c25 100644
--- a/vtkm/exec/arg/FetchTagKeysIn.h
+++ b/vtkm/exec/arg/FetchTagKeysIn.h
@@ -36,24 +36,22 @@ template <typename KeyPortalType, typename IdPortalType, typename IdComponentPor
 struct Fetch<
   vtkm::exec::arg::FetchTagKeysIn,
   vtkm::exec::arg::AspectTagDefault,
-  vtkm::exec::arg::ThreadIndicesReduceByKey,
   vtkm::exec::internal::ReduceByKeyLookup<KeyPortalType, IdPortalType, IdComponentPortalType>>
 {
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesReduceByKey;
   using ExecObjectType =
     vtkm::exec::internal::ReduceByKeyLookup<KeyPortalType, IdPortalType, IdComponentPortalType>;
 
   using ValueType = typename ExecObjectType::KeyType;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType& keys) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices, const ExecObjectType& keys) const
   {
     return keys.UniqueKeys.Get(indices.GetInputIndex());
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op for this fetch.
   }
diff --git a/vtkm/exec/arg/FetchTagWholeCellSetIn.h b/vtkm/exec/arg/FetchTagWholeCellSetIn.h
index b57a00655..59a456d1f 100644
--- a/vtkm/exec/arg/FetchTagWholeCellSetIn.h
+++ b/vtkm/exec/arg/FetchTagWholeCellSetIn.h
@@ -29,24 +29,23 @@ struct FetchTagWholeCellSetIn
 {
 };
 
-template <typename ThreadIndicesType, typename ExecObjectType>
+template <typename ExecObjectType>
 struct Fetch<vtkm::exec::arg::FetchTagWholeCellSetIn,
              vtkm::exec::arg::AspectTagDefault,
-             ThreadIndicesType,
              ExecObjectType>
 {
   using ValueType = ExecObjectType;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& vtkmNotUsed(indices),
-                 const ExecObjectType& execObject) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& vtkmNotUsed(indices),
+                           const ExecObjectType& execObject) const
   {
     return execObject;
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op for this fetch.
   }
diff --git a/vtkm/exec/arg/IncidentElementCount.h b/vtkm/exec/arg/IncidentElementCount.h
index 251467bf9..1ca276adc 100644
--- a/vtkm/exec/arg/IncidentElementCount.h
+++ b/vtkm/exec/arg/IncidentElementCount.h
@@ -12,7 +12,6 @@
 
 #include <vtkm/exec/arg/ExecutionSignatureTagBase.h>
 #include <vtkm/exec/arg/Fetch.h>
-#include <vtkm/exec/arg/ThreadIndicesTopologyMap.h>
 
 namespace vtkm
 {
@@ -44,25 +43,20 @@ struct IncidentElementCount : vtkm::exec::arg::ExecutionSignatureTagBase
   using AspectTag = vtkm::exec::arg::AspectTagIncidentElementCount;
 };
 
-template <typename FetchTag, typename ConnectivityType, typename ExecObjectType>
-struct Fetch<FetchTag,
-             vtkm::exec::arg::AspectTagIncidentElementCount,
-             vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>,
-             ExecObjectType>
+template <typename FetchTag, typename ExecObjectType>
+struct Fetch<FetchTag, vtkm::exec::arg::AspectTagIncidentElementCount, ExecObjectType>
 {
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>;
-
   using ValueType = vtkm::IdComponent;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices, const ExecObjectType&) const
   {
     return indices.GetIndicesIncident().GetNumberOfComponents();
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op.
   }
diff --git a/vtkm/exec/arg/IncidentElementIndices.h b/vtkm/exec/arg/IncidentElementIndices.h
index f14cc81bd..d90b930f6 100644
--- a/vtkm/exec/arg/IncidentElementIndices.h
+++ b/vtkm/exec/arg/IncidentElementIndices.h
@@ -12,7 +12,6 @@
 
 #include <vtkm/exec/arg/ExecutionSignatureTagBase.h>
 #include <vtkm/exec/arg/Fetch.h>
-#include <vtkm/exec/arg/ThreadIndicesTopologyMap.h>
 
 namespace vtkm
 {
@@ -44,30 +43,6 @@ struct IncidentElementIndices : vtkm::exec::arg::ExecutionSignatureTagBase
   static constexpr vtkm::IdComponent INDEX = 1;
   using AspectTag = vtkm::exec::arg::AspectTagIncidentElementIndices;
 };
-
-template <typename FetchTag, typename ConnectivityType, typename ExecObjectType>
-struct Fetch<FetchTag,
-             vtkm::exec::arg::AspectTagIncidentElementIndices,
-             vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>,
-             ExecObjectType>
-{
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesTopologyMap<ConnectivityType>;
-
-  using ValueType = typename ThreadIndicesType::IndicesIncidentType;
-
-  VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType&) const
-  {
-    return indices.GetIndicesIncident();
-  }
-
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
-  {
-    // Store is a no-op.
-  }
-};
 }
 }
 } // namespace vtkm::exec::arg
diff --git a/vtkm/exec/arg/InputIndex.h b/vtkm/exec/arg/InputIndex.h
index ffa2de996..c71dd3eb2 100644
--- a/vtkm/exec/arg/InputIndex.h
+++ b/vtkm/exec/arg/InputIndex.h
@@ -50,19 +50,19 @@ struct InputIndex : vtkm::exec::arg::ExecutionSignatureTagBase
   using AspectTag = vtkm::exec::arg::AspectTagInputIndex;
 };
 
-template <typename FetchTag, typename ThreadIndicesType, typename ExecObjectType>
-struct Fetch<FetchTag, vtkm::exec::arg::AspectTagInputIndex, ThreadIndicesType, ExecObjectType>
+template <typename FetchTag, typename ExecObjectType>
+struct Fetch<FetchTag, vtkm::exec::arg::AspectTagInputIndex, ExecObjectType>
 {
   using ValueType = vtkm::Id;
 
-  VTKM_EXEC
-  vtkm::Id Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC vtkm::Id Load(const ThreadIndicesType& indices, const ExecObjectType&) const
   {
     return indices.GetInputIndex();
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op.
   }
diff --git a/vtkm/exec/arg/OutputIndex.h b/vtkm/exec/arg/OutputIndex.h
index 665ba00a1..1d426f57d 100644
--- a/vtkm/exec/arg/OutputIndex.h
+++ b/vtkm/exec/arg/OutputIndex.h
@@ -50,19 +50,19 @@ struct OutputIndex : vtkm::exec::arg::ExecutionSignatureTagBase
   using AspectTag = vtkm::exec::arg::AspectTagOutputIndex;
 };
 
-template <typename FetchTag, typename ThreadIndicesType, typename ExecObjectType>
-struct Fetch<FetchTag, vtkm::exec::arg::AspectTagOutputIndex, ThreadIndicesType, ExecObjectType>
+template <typename FetchTag, typename ExecObjectType>
+struct Fetch<FetchTag, vtkm::exec::arg::AspectTagOutputIndex, ExecObjectType>
 {
   using ValueType = vtkm::Id;
 
-  VTKM_EXEC
-  vtkm::Id Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC vtkm::Id Load(const ThreadIndicesType& indices, const ExecObjectType&) const
   {
     return indices.GetOutputIndex();
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op.
   }
diff --git a/vtkm/exec/arg/ThreadIndices.h b/vtkm/exec/arg/ThreadIndices.h
index e000d3d3c..90f6be812 100644
--- a/vtkm/exec/arg/ThreadIndices.h
+++ b/vtkm/exec/arg/ThreadIndices.h
@@ -49,19 +49,21 @@ struct ThreadIndices : vtkm::exec::arg::ExecutionSignatureTagBase
   using AspectTag = vtkm::exec::arg::AspectTagThreadIndices;
 };
 
-template <typename FetchTag, typename ThreadIndicesType, typename ExecObjectType>
-struct Fetch<FetchTag, vtkm::exec::arg::AspectTagThreadIndices, ThreadIndicesType, ExecObjectType>
+template <typename FetchTag, typename ExecObjectType>
+struct Fetch<FetchTag, vtkm::exec::arg::AspectTagThreadIndices, ExecObjectType>
 {
-  using ValueType = const ThreadIndicesType&;
 
-  VTKM_EXEC
-  const ThreadIndicesType& Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC const ThreadIndicesType& Load(const ThreadIndicesType& indices,
+                                          const ExecObjectType&) const
   {
     return indices;
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ThreadIndicesType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&,
+                       const ExecObjectType&,
+                       const ThreadIndicesType&) const
   {
     // Store is a no-op.
   }
diff --git a/vtkm/exec/arg/ThreadIndicesExtrude.h b/vtkm/exec/arg/ThreadIndicesExtrude.h
index 3db0498e6..21f9f1f5c 100644
--- a/vtkm/exec/arg/ThreadIndicesExtrude.h
+++ b/vtkm/exec/arg/ThreadIndicesExtrude.h
@@ -31,7 +31,7 @@ public:
   using CellShapeTag = typename ConnectivityType::CellShapeTag;
   using IndicesIncidentType = typename ConnectivityType::IndicesType;
   using LogicalIndexType = typename ConnectivityType::SchedulingRangeType;
-
+  using Connectivity = vtkm::exec::ConnectivityExtrude<Device>;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
   VTKM_EXEC ThreadIndicesTopologyMap(vtkm::Id threadIndex,
@@ -184,6 +184,7 @@ public:
   using CellShapeTag = typename ConnectivityType::CellShapeTag;
   using IndicesIncidentType = typename ConnectivityType::IndicesType;
   using LogicalIndexType = typename ConnectivityType::SchedulingRangeType;
+  using Connectivity = ConnectivityType;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
   VTKM_EXEC
diff --git a/vtkm/exec/arg/ThreadIndicesTopologyMap.h b/vtkm/exec/arg/ThreadIndicesTopologyMap.h
index 8809365be..a32f85a0b 100644
--- a/vtkm/exec/arg/ThreadIndicesTopologyMap.h
+++ b/vtkm/exec/arg/ThreadIndicesTopologyMap.h
@@ -88,6 +88,7 @@ class ThreadIndicesTopologyMap : public vtkm::exec::arg::ThreadIndicesBasic
 public:
   using IndicesIncidentType = typename ConnectivityType::IndicesType;
   using CellShapeTag = typename ConnectivityType::CellShapeTag;
+  using Connectivity = ConnectivityType;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
   VTKM_EXEC ThreadIndicesTopologyMap(vtkm::Id threadIndex,
@@ -155,6 +156,8 @@ public:
   using IndicesIncidentType = typename ConnectivityType::IndicesType;
   using CellShapeTag = typename ConnectivityType::CellShapeTag;
   using LogicalIndexType = typename ConnectivityType::SchedulingRangeType;
+  using Connectivity =
+    vtkm::exec::ConnectivityStructured<VisitTopology, IncidentTopology, Dimension>;
 
   VTKM_EXEC ThreadIndicesTopologyMap(vtkm::Id threadIndex,
                                      vtkm::Id inIndex,
@@ -317,6 +320,11 @@ public:
   using IndicesIncidentType = typename ConnectivityType::IndicesType;
   using CellShapeTag = typename ConnectivityType::CellShapeTag;
   using LogicalIndexType = typename ConnectivityType::SchedulingRangeType;
+  using Connectivity = vtkm::exec::ConnectivityPermutedVisitCellsWithPoints<
+    PermutationPortal,
+    vtkm::exec::ConnectivityStructured<vtkm::TopologyElementTagCell,
+                                       vtkm::TopologyElementTagPoint,
+                                       Dimension>>;
 
   VTKM_EXEC ThreadIndicesTopologyMap(vtkm::Id threadIndex,
                                      vtkm::Id inputIndex,
diff --git a/vtkm/exec/arg/ValueCount.h b/vtkm/exec/arg/ValueCount.h
index 9caf6860e..e97af9966 100644
--- a/vtkm/exec/arg/ValueCount.h
+++ b/vtkm/exec/arg/ValueCount.h
@@ -44,24 +44,20 @@ struct ValueCount : vtkm::exec::arg::ExecutionSignatureTagBase
 };
 
 template <typename FetchTag, typename ExecObjectType>
-struct Fetch<FetchTag,
-             vtkm::exec::arg::AspectTagValueCount,
-             vtkm::exec::arg::ThreadIndicesReduceByKey,
-             ExecObjectType>
+struct Fetch<FetchTag, vtkm::exec::arg::AspectTagValueCount, ExecObjectType>
 {
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesReduceByKey;
 
   using ValueType = vtkm::IdComponent;
 
   VTKM_SUPPRESS_EXEC_WARNINGS
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices, const ExecObjectType&) const
   {
     return indices.GetNumberOfValues();
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op.
   }
diff --git a/vtkm/exec/arg/VisitIndex.h b/vtkm/exec/arg/VisitIndex.h
index 1d055713b..7be61cf7e 100644
--- a/vtkm/exec/arg/VisitIndex.h
+++ b/vtkm/exec/arg/VisitIndex.h
@@ -51,19 +51,19 @@ struct VisitIndex : vtkm::exec::arg::ExecutionSignatureTagBase
   using AspectTag = vtkm::exec::arg::AspectTagVisitIndex;
 };
 
-template <typename FetchTag, typename ThreadIndicesType, typename ExecObjectType>
-struct Fetch<FetchTag, vtkm::exec::arg::AspectTagVisitIndex, ThreadIndicesType, ExecObjectType>
+template <typename FetchTag, typename ExecObjectType>
+struct Fetch<FetchTag, vtkm::exec::arg::AspectTagVisitIndex, ExecObjectType>
 {
   using ValueType = vtkm::IdComponent;
 
-  VTKM_EXEC
-  vtkm::IdComponent Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC vtkm::IdComponent Load(const ThreadIndicesType& indices, const ExecObjectType&) const
   {
     return indices.GetVisitIndex();
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op.
   }
diff --git a/vtkm/exec/arg/WorkIndex.h b/vtkm/exec/arg/WorkIndex.h
index d4af78062..c4799a5f6 100644
--- a/vtkm/exec/arg/WorkIndex.h
+++ b/vtkm/exec/arg/WorkIndex.h
@@ -47,19 +47,19 @@ struct WorkIndex : vtkm::exec::arg::ExecutionSignatureTagBase
   using AspectTag = vtkm::exec::arg::AspectTagWorkIndex;
 };
 
-template <typename FetchTag, typename ThreadIndicesType, typename ExecObjectType>
-struct Fetch<FetchTag, vtkm::exec::arg::AspectTagWorkIndex, ThreadIndicesType, ExecObjectType>
+template <typename FetchTag, typename ExecObjectType>
+struct Fetch<FetchTag, vtkm::exec::arg::AspectTagWorkIndex, ExecObjectType>
 {
   using ValueType = vtkm::Id;
 
-  VTKM_EXEC
-  vtkm::Id Load(const ThreadIndicesType& indices, const ExecObjectType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC vtkm::Id Load(const ThreadIndicesType& indices, const ExecObjectType&) const
   {
     return indices.GetThreadIndex();
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const ExecObjectType&, const ValueType&) const
   {
     // Store is a no-op.
   }
diff --git a/vtkm/exec/arg/testing/UnitTestFetchArrayDirectIn.cxx b/vtkm/exec/arg/testing/UnitTestFetchArrayDirectIn.cxx
index 2bed5a70f..0d638ce31 100644
--- a/vtkm/exec/arg/testing/UnitTestFetchArrayDirectIn.cxx
+++ b/vtkm/exec/arg/testing/UnitTestFetchArrayDirectIn.cxx
@@ -45,7 +45,6 @@ struct FetchArrayDirectInTests
 
     using FetchType = vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayDirectIn,
                                              vtkm::exec::arg::AspectTagDefault,
-                                             vtkm::exec::arg::ThreadIndicesTesting,
                                              TestPortal<T>>;
 
     FetchType fetch;
diff --git a/vtkm/exec/arg/testing/UnitTestFetchArrayDirectIn3d.cxx b/vtkm/exec/arg/testing/UnitTestFetchArrayDirectIn3d.cxx
index fea0dd9d3..ad21f53cd 100644
--- a/vtkm/exec/arg/testing/UnitTestFetchArrayDirectIn3d.cxx
+++ b/vtkm/exec/arg/testing/UnitTestFetchArrayDirectIn3d.cxx
@@ -54,21 +54,21 @@ namespace arg
 template <typename T>
 struct Fetch<vtkm::exec::arg::FetchTagArrayDirectIn,
              vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic3D,
              TestPortal<T>>
 {
   using ValueType = T;
   using PortalType = const TestPortal<T>&;
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesBasic3D;
 
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, PortalType field) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices, PortalType field) const
   {
     return field.Get(indices.GetInputIndex3D());
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, PortalType, ValueType) const {}
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, PortalType, ValueType) const
+  {
+  }
 };
 }
 }
@@ -86,7 +86,6 @@ struct FetchArrayDirectIn3DTests
 
     using FetchType = vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayDirectIn,
                                              vtkm::exec::arg::AspectTagDefault,
-                                             vtkm::exec::arg::ThreadIndicesBasic3D,
                                              TestPortal<T>>;
 
     FetchType fetch;
diff --git a/vtkm/exec/arg/testing/UnitTestFetchArrayDirectInOut.cxx b/vtkm/exec/arg/testing/UnitTestFetchArrayDirectInOut.cxx
index bfc094e41..55f358977 100644
--- a/vtkm/exec/arg/testing/UnitTestFetchArrayDirectInOut.cxx
+++ b/vtkm/exec/arg/testing/UnitTestFetchArrayDirectInOut.cxx
@@ -58,7 +58,6 @@ struct FetchArrayDirectInTests
 
     using FetchType = vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayDirectInOut,
                                              vtkm::exec::arg::AspectTagDefault,
-                                             vtkm::exec::arg::ThreadIndicesTesting,
                                              TestPortal<T>>;
 
     FetchType fetch;
diff --git a/vtkm/exec/arg/testing/UnitTestFetchArrayDirectOut.cxx b/vtkm/exec/arg/testing/UnitTestFetchArrayDirectOut.cxx
index f714ea75b..d829b6a71 100644
--- a/vtkm/exec/arg/testing/UnitTestFetchArrayDirectOut.cxx
+++ b/vtkm/exec/arg/testing/UnitTestFetchArrayDirectOut.cxx
@@ -50,7 +50,6 @@ struct FetchArrayDirectOutTests
 
     using FetchType = vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayDirectOut,
                                              vtkm::exec::arg::AspectTagDefault,
-                                             vtkm::exec::arg::ThreadIndicesTesting,
                                              TestPortal<T>>;
 
     FetchType fetch;
diff --git a/vtkm/exec/arg/testing/UnitTestFetchArrayNeighborhoodIn.cxx b/vtkm/exec/arg/testing/UnitTestFetchArrayNeighborhoodIn.cxx
index e824f9fa2..99b970dab 100644
--- a/vtkm/exec/arg/testing/UnitTestFetchArrayNeighborhoodIn.cxx
+++ b/vtkm/exec/arg/testing/UnitTestFetchArrayNeighborhoodIn.cxx
@@ -96,7 +96,6 @@ struct FetchArrayNeighborhoodInTests
 
     using FetchType = vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayNeighborhoodIn,
                                              vtkm::exec::arg::AspectTagDefault,
-                                             vtkm::exec::arg::ThreadIndicesPointNeighborhood,
                                              TestPortal<T>>;
 
     FetchType fetch;
diff --git a/vtkm/exec/arg/testing/UnitTestFetchArrayTopologyMapIn.cxx b/vtkm/exec/arg/testing/UnitTestFetchArrayTopologyMapIn.cxx
index 5ef6873cc..1ab639ed1 100644
--- a/vtkm/exec/arg/testing/UnitTestFetchArrayTopologyMapIn.cxx
+++ b/vtkm/exec/arg/testing/UnitTestFetchArrayTopologyMapIn.cxx
@@ -8,15 +8,13 @@
 //  PURPOSE.  See the above copyright notice for more information.
 //============================================================================
 
+#include <vtkm/testing/Testing.h>
+
 #include <vtkm/exec/arg/FetchTagArrayTopologyMapIn.h>
-
 #include <vtkm/exec/arg/ThreadIndicesTopologyMap.h>
-
 #include <vtkm/internal/FunctionInterface.h>
 #include <vtkm/internal/Invocation.h>
 
-#include <vtkm/testing/Testing.h>
-
 namespace
 {
 
@@ -93,7 +91,6 @@ struct FetchArrayTopologyMapInTests
 
     using FetchType = vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayTopologyMapIn,
                                              vtkm::exec::arg::AspectTagDefault,
-                                             ThreadIndicesType,
                                              TestPortal<T>>;
 
     FetchType fetch;
@@ -105,7 +102,7 @@ struct FetchArrayTopologyMapInTests
     ThreadIndicesType indices(
       threadIndex, inputIndex, visitIndex, outputIndex, invocation.GetInputDomain());
 
-    typename FetchType::ValueType value =
+    auto value =
       fetch.Load(indices, vtkm::internal::ParameterGet<ParamIndex>(invocation.Parameters));
     VTKM_TEST_ASSERT(value.GetNumberOfComponents() == 8,
                      "Topology fetch got wrong number of components.");
@@ -173,7 +170,6 @@ void TryStructuredPointCoordinatesInvocation(const Invocation& invocation)
 
   vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayTopologyMapIn,
                          vtkm::exec::arg::AspectTagDefault,
-                         ThreadIndicesType,
                          vtkm::internal::ArrayPortalUniformPointCoordinates>
     fetch;
 
diff --git a/vtkm/exec/arg/testing/UnitTestFetchExecObject.cxx b/vtkm/exec/arg/testing/UnitTestFetchExecObject.cxx
index 651224ea1..3df0af672 100644
--- a/vtkm/exec/arg/testing/UnitTestFetchExecObject.cxx
+++ b/vtkm/exec/arg/testing/UnitTestFetchExecObject.cxx
@@ -38,7 +38,6 @@ void TryInvocation()
 
   using FetchType = vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagExecObject,
                                            vtkm::exec::arg::AspectTagDefault,
-                                           vtkm::exec::arg::ThreadIndicesTesting,
                                            TestExecutionObject>;
 
   FetchType fetch;
diff --git a/vtkm/exec/arg/testing/UnitTestFetchWorkIndex.cxx b/vtkm/exec/arg/testing/UnitTestFetchWorkIndex.cxx
index 9df7f1802..c23ad40b8 100644
--- a/vtkm/exec/arg/testing/UnitTestFetchWorkIndex.cxx
+++ b/vtkm/exec/arg/testing/UnitTestFetchWorkIndex.cxx
@@ -26,7 +26,6 @@ void TestWorkIndexFetch()
   using FetchType =
     vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayDirectIn, // Not used but probably common.
                            vtkm::exec::arg::AspectTagWorkIndex,
-                           vtkm::exec::arg::ThreadIndicesTesting,
                            vtkm::internal::NullType>;
 
   FetchType fetch;
diff --git a/vtkm/exec/cuda/internal/testing/UnitTestTaskStrided.cu b/vtkm/exec/cuda/internal/testing/UnitTestTaskStrided.cu
index e43f47a15..c68db0c92 100644
--- a/vtkm/exec/cuda/internal/testing/UnitTestTaskStrided.cu
+++ b/vtkm/exec/cuda/internal/testing/UnitTestTaskStrided.cu
@@ -97,10 +97,7 @@ namespace arg
 {
 
 template <>
-struct Fetch<TestFetchTagInput,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic,
-             TestExecObject>
+struct Fetch<TestFetchTagInput, vtkm::exec::arg::AspectTagDefault, TestExecObject>
 {
   using ValueType = vtkm::Id;
 
@@ -119,10 +116,7 @@ struct Fetch<TestFetchTagInput,
 };
 
 template <>
-struct Fetch<TestFetchTagOutput,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic,
-             TestExecObject>
+struct Fetch<TestFetchTagOutput, vtkm::exec::arg::AspectTagDefault, TestExecObject>
 {
   using ValueType = vtkm::Id;
 
diff --git a/vtkm/exec/internal/WorkletInvokeFunctorDetail.h b/vtkm/exec/internal/WorkletInvokeFunctorDetail.h
index 178e789fa..2d3f6229d 100644
--- a/vtkm/exec/internal/WorkletInvokeFunctorDetail.h
+++ b/vtkm/exec/internal/WorkletInvokeFunctorDetail.h
@@ -84,7 +84,7 @@ struct InvocationToFetch
       typename Invocation::DeviceAdapterTag,
       typename Invocation::ParameterInterface::template ParameterType<ControlParameterIndex>::type>::type;
 
-  using type = vtkm::exec::arg::Fetch<FetchTag, AspectTag, ThreadIndicesType, ExecObjectType>;
+  using type = vtkm::exec::arg::Fetch<FetchTag, AspectTag, ExecObjectType>;
 
   VTKM_EXEC static ExecObjectType GetParameterImpl(const Invocation&, std::true_type)
   {
diff --git a/vtkm/exec/internal/WorkletInvokeFunctorDetail.h.in b/vtkm/exec/internal/WorkletInvokeFunctorDetail.h.in
index af3548c32..679043a63 100644
--- a/vtkm/exec/internal/WorkletInvokeFunctorDetail.h.in
+++ b/vtkm/exec/internal/WorkletInvokeFunctorDetail.h.in
@@ -145,7 +145,7 @@ struct InvocationToFetch
       typename Invocation::DeviceAdapterTag,
       typename Invocation::ParameterInterface::template ParameterType<ControlParameterIndex>::type>::type;
 
-  using type = vtkm::exec::arg::Fetch<FetchTag, AspectTag, ThreadIndicesType, ExecObjectType>;
+  using type = vtkm::exec::arg::Fetch<FetchTag, AspectTag, ExecObjectType>;
 
   VTKM_EXEC static ExecObjectType GetParameterImpl(const Invocation&, std::true_type)
   {
diff --git a/vtkm/exec/internal/testing/TestingTaskTiling.h b/vtkm/exec/internal/testing/TestingTaskTiling.h
index c7d4152d3..0a02f83df 100644
--- a/vtkm/exec/internal/testing/TestingTaskTiling.h
+++ b/vtkm/exec/internal/testing/TestingTaskTiling.h
@@ -104,10 +104,7 @@ namespace arg
 using namespace vtkm::exec::internal::testing;
 
 template <>
-struct Fetch<TestFetchTagInput,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic,
-             TestExecObject>
+struct Fetch<TestFetchTagInput, vtkm::exec::arg::AspectTagDefault, TestExecObject>
 {
   using ValueType = vtkm::Id;
 
@@ -126,10 +123,7 @@ struct Fetch<TestFetchTagInput,
 };
 
 template <>
-struct Fetch<TestFetchTagOutput,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic,
-             TestExecObject>
+struct Fetch<TestFetchTagOutput, vtkm::exec::arg::AspectTagDefault, TestExecObject>
 {
   using ValueType = vtkm::Id;
 
diff --git a/vtkm/exec/internal/testing/UnitTestTaskSingular.cxx b/vtkm/exec/internal/testing/UnitTestTaskSingular.cxx
index d0144fa52..c1940fa62 100644
--- a/vtkm/exec/internal/testing/UnitTestTaskSingular.cxx
+++ b/vtkm/exec/internal/testing/UnitTestTaskSingular.cxx
@@ -87,10 +87,7 @@ namespace arg
 {
 
 template <>
-struct Fetch<TestFetchTagInput,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic,
-             TestExecObject>
+struct Fetch<TestFetchTagInput, vtkm::exec::arg::AspectTagDefault, TestExecObject>
 {
   using ValueType = vtkm::Id;
 
@@ -109,10 +106,7 @@ struct Fetch<TestFetchTagInput,
 };
 
 template <>
-struct Fetch<TestFetchTagOutput,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic,
-             TestExecObject>
+struct Fetch<TestFetchTagOutput, vtkm::exec::arg::AspectTagDefault, TestExecObject>
 {
   using ValueType = vtkm::Id;
 
@@ -226,7 +220,6 @@ VTKM_STATIC_ASSERT(
                   InvocationToFetch<vtkm::exec::arg::ThreadIndicesBasic, InvocationType1, 1>::type,
                 vtkm::exec::arg::Fetch<TestFetchTagInput,
                                        vtkm::exec::arg::AspectTagDefault,
-                                       vtkm::exec::arg::ThreadIndicesBasic,
                                        TestExecObject>>::type::value));
 
 VTKM_STATIC_ASSERT(
@@ -234,7 +227,6 @@ VTKM_STATIC_ASSERT(
                   InvocationToFetch<vtkm::exec::arg::ThreadIndicesBasic, InvocationType1, 2>::type,
                 vtkm::exec::arg::Fetch<TestFetchTagOutput,
                                        vtkm::exec::arg::AspectTagDefault,
-                                       vtkm::exec::arg::ThreadIndicesBasic,
                                        TestExecObject>>::type::value));
 
 VTKM_STATIC_ASSERT(
@@ -242,7 +234,6 @@ VTKM_STATIC_ASSERT(
                   InvocationToFetch<vtkm::exec::arg::ThreadIndicesBasic, InvocationType2, 0>::type,
                 vtkm::exec::arg::Fetch<TestFetchTagOutput,
                                        vtkm::exec::arg::AspectTagDefault,
-                                       vtkm::exec::arg::ThreadIndicesBasic,
                                        TestExecObject>>::type::value));
 
 void TestNormalFunctorInvoke()
diff --git a/vtkm/exec/internal/testing/UnitTestWorkletInvokeFunctor.cxx b/vtkm/exec/internal/testing/UnitTestWorkletInvokeFunctor.cxx
index 9113aee4a..df6f22974 100644
--- a/vtkm/exec/internal/testing/UnitTestWorkletInvokeFunctor.cxx
+++ b/vtkm/exec/internal/testing/UnitTestWorkletInvokeFunctor.cxx
@@ -88,10 +88,7 @@ namespace arg
 {
 
 template <>
-struct Fetch<TestFetchTagInput,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic,
-             TestExecObject>
+struct Fetch<TestFetchTagInput, vtkm::exec::arg::AspectTagDefault, TestExecObject>
 {
   using ValueType = vtkm::Id;
 
@@ -110,10 +107,7 @@ struct Fetch<TestFetchTagInput,
 };
 
 template <>
-struct Fetch<TestFetchTagOutput,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic,
-             TestExecObject>
+struct Fetch<TestFetchTagOutput, vtkm::exec::arg::AspectTagDefault, TestExecObject>
 {
   using ValueType = vtkm::Id;
 
diff --git a/vtkm/worklet/gradient/PointGradient.h b/vtkm/worklet/gradient/PointGradient.h
index e77322bae..8960c569c 100644
--- a/vtkm/worklet/gradient/PointGradient.h
+++ b/vtkm/worklet/gradient/PointGradient.h
@@ -15,6 +15,7 @@
 #include <vtkm/exec/ParametricCoordinates.h>
 #include <vtkm/worklet/WorkletMapTopology.h>
 
+#include <utility>
 #include <vtkm/worklet/gradient/GradientOutput.h>
 
 
@@ -50,6 +51,7 @@ struct PointGradient : public vtkm::worklet::WorkletVisitPointsWithCells
                             GradientOutType& outputGradient) const
   {
     using CellThreadIndices = vtkm::exec::arg::ThreadIndicesTopologyMap<CellSetInType>;
+
     using ValueType = typename WholeFieldIn::ValueType;
     using CellShapeTag = typename CellSetInType::CellShapeTag;
 
@@ -107,10 +109,9 @@ private:
     }
   }
 
-  template <typename CellSetInType>
-  VTKM_EXEC vtkm::IdComponent GetPointIndexForCell(
-    const vtkm::exec::arg::ThreadIndicesTopologyMap<CellSetInType>& indices,
-    vtkm::Id pointId) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC vtkm::IdComponent GetPointIndexForCell(const ThreadIndicesType& indices,
+                                                   vtkm::Id pointId) const
   {
     vtkm::IdComponent result = 0;
     const auto& topo = indices.GetIndicesIncident();
@@ -128,14 +129,12 @@ private:
   //VecRectilinearPointCoordinates when using structured connectivity, and
   //uniform point coordinates.
   //c++14 would make the return type simply auto
-  template <typename CellSetInType, typename WholeFieldIn>
-  VTKM_EXEC
-    typename vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayTopologyMapIn,
-                                    vtkm::exec::arg::AspectTagDefault,
-                                    vtkm::exec::arg::ThreadIndicesTopologyMap<CellSetInType>,
-                                    typename WholeFieldIn::PortalType>::ValueType
-    GetValues(const vtkm::exec::arg::ThreadIndicesTopologyMap<CellSetInType>& indices,
-              const WholeFieldIn& in) const
+  template <typename ThreadIndicesType, typename WholeFieldIn>
+  VTKM_EXEC auto GetValues(const ThreadIndicesType& indices, const WholeFieldIn& in) const
+    -> decltype(std::declval<vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayTopologyMapIn,
+                                                    vtkm::exec::arg::AspectTagDefault,
+                                                    typename WholeFieldIn::PortalType>>()
+                  .Load(indices, in.GetPortal()))
   {
     //the current problem is that when the topology is structured
     //we are passing in an vtkm::Id when it wants a Id2 or an Id3 that
@@ -143,7 +142,6 @@ private:
     using ExecObjectType = typename WholeFieldIn::PortalType;
     using Fetch = vtkm::exec::arg::Fetch<vtkm::exec::arg::FetchTagArrayTopologyMapIn,
                                          vtkm::exec::arg::AspectTagDefault,
-                                         vtkm::exec::arg::ThreadIndicesTopologyMap<CellSetInType>,
                                          ExecObjectType>;
     Fetch fetch;
     return fetch.Load(indices, in.GetPortal());
diff --git a/vtkm/worklet/internal/testing/UnitTestDispatcherBase.cxx b/vtkm/worklet/internal/testing/UnitTestDispatcherBase.cxx
index 373c57365..0acecaaf5 100644
--- a/vtkm/worklet/internal/testing/UnitTestDispatcherBase.cxx
+++ b/vtkm/worklet/internal/testing/UnitTestDispatcherBase.cxx
@@ -185,10 +185,7 @@ namespace arg
 {
 
 template <>
-struct Fetch<TestFetchTagInput,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic,
-             TestExecObjectIn>
+struct Fetch<TestFetchTagInput, vtkm::exec::arg::AspectTagDefault, TestExecObjectIn>
 {
   using ValueType = vtkm::Id;
 
@@ -207,10 +204,7 @@ struct Fetch<TestFetchTagInput,
 };
 
 template <>
-struct Fetch<TestFetchTagOutput,
-             vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic,
-             TestExecObjectOut>
+struct Fetch<TestFetchTagOutput, vtkm::exec::arg::AspectTagDefault, TestExecObjectOut>
 {
   using ValueType = vtkm::Id;
 
diff --git a/vtkm/worklet/testing/UnitTestWorkletMapField3d.cxx b/vtkm/worklet/testing/UnitTestWorkletMapField3d.cxx
index 272517fc6..aeda0a6f4 100644
--- a/vtkm/worklet/testing/UnitTestWorkletMapField3d.cxx
+++ b/vtkm/worklet/testing/UnitTestWorkletMapField3d.cxx
@@ -59,21 +59,21 @@ namespace arg
 template <typename PType>
 struct Fetch<vtkm::exec::arg::FetchTagExecObject,
              vtkm::exec::arg::AspectTagDefault,
-             vtkm::exec::arg::ThreadIndicesBasic3D,
              mapfield3d::ExecutionObject<PType>>
 {
   using ValueType = typename PType::ValueType;
   using PortalType = mapfield3d::ExecutionObject<PType>;
-  using ThreadIndicesType = vtkm::exec::arg::ThreadIndicesBasic3D;
 
-  VTKM_EXEC
-  ValueType Load(const ThreadIndicesType& indices, const PortalType& field) const
+  template <typename ThreadIndicesType>
+  VTKM_EXEC ValueType Load(const ThreadIndicesType& indices, const PortalType& field) const
   {
     return field.Portal.Get(indices.GetInputIndex());
   }
 
-  VTKM_EXEC
-  void Store(const ThreadIndicesType&, const PortalType&, const ValueType&) const {}
+  template <typename ThreadIndicesType>
+  VTKM_EXEC void Store(const ThreadIndicesType&, const PortalType&, const ValueType&) const
+  {
+  }
 };
 }
 }