diff --git a/mercurial/branchmap.py b/mercurial/branchmap.py
--- a/mercurial/branchmap.py
+++ b/mercurial/branchmap.py
@@ -62,6 +62,7 @@ class BranchMapCache:
     def __getitem__(self, repo):
         self.updatecache(repo)
         bcache = self._per_filter[repo.filtername]
+        bcache._ensure_populated(repo)
         assert bcache._filtername == repo.filtername, (
             bcache._filtername,
             repo.filtername,
@@ -484,6 +485,9 @@ class _LocalBranchCache(_BaseBranchCache
     def _compute_key_hashes(self, repo) -> Tuple[bytes]:
         raise NotImplementedError
 
+    def _ensure_populated(self, repo):
+        """make sure any lazily loaded values are fully populated"""
+
     def validfor(self, repo):
         """check that cache contents are valid for (a subset of) this repo
 
@@ -861,7 +865,18 @@ class BranchCacheV3(_LocalBranchCache):
     _base_filename = b"branch3"
     _default_key_hashes = (None, None)
 
-    def _get_topo_heads(self, repo) -> List[int]:
+    def __init__(self, *args, pure_topo_branch=None, **kwargs):
+        super().__init__(*args, **kwargs)
+        self._pure_topo_branch = pure_topo_branch
+        self._needs_populate = self._pure_topo_branch is not None
+
+    def inherit_for(self, repo):
+        new = super().inherit_for(repo)
+        new._pure_topo_branch = self._pure_topo_branch
+        new._needs_populate = self._needs_populate
+        return new
+
+    def _get_topo_heads(self, repo):
         """returns the topological head of a repoview content up to self.tiprev"""
         cl = repo.changelog
         if self.tiprev == nullrev:
@@ -883,22 +898,33 @@ class BranchCacheV3(_LocalBranchCache):
                 cache_keys[b"filtered-hash"] = hex(self.key_hashes[0])
             if self.key_hashes[1] is not None:
                 cache_keys[b"obsolete-hash"] = hex(self.key_hashes[1])
+        if self._pure_topo_branch is not None:
+            cache_keys[b"topo-mode"] = b"pure"
         pieces = (b"%s=%s" % i for i in sorted(cache_keys.items()))
         fp.write(b" ".join(pieces) + b'\n')
+        if self._pure_topo_branch is not None:
+            label = encoding.fromlocal(self._pure_topo_branch)
+            fp.write(label + b'\n')
 
     def _write_heads(self, repo, fp) -> int:
         """write list of heads to a file
 
         Return the number of heads written."""
         nodecount = 0
-        topo_heads = set(self._get_topo_heads(repo))
+        topo_heads = None
+        if self._pure_topo_branch is None:
+            topo_heads = set(self._get_topo_heads(repo))
         to_rev = repo.changelog.index.rev
         for label, nodes in sorted(self._entries.items()):
+            if label == self._pure_topo_branch:
+                # not need to write anything the header took care of that
+                continue
             label = encoding.fromlocal(label)
             for node in nodes:
-                rev = to_rev(node)
-                if rev in topo_heads:
-                    continue
+                if topo_heads is not None:
+                    rev = to_rev(node)
+                    if rev in topo_heads:
+                        continue
                 if node in self._closednodes:
                     state = b'c'
                 else:
@@ -916,6 +942,7 @@ class BranchCacheV3(_LocalBranchCache):
         args = {}
         filtered_hash = None
         obsolete_hash = None
+        has_pure_topo_heads = False
         for k, v in cache_keys.items():
             if k == b"tip-rev":
                 args["tiprev"] = int(v)
@@ -925,16 +952,28 @@ class BranchCacheV3(_LocalBranchCache):
                 filtered_hash = bin(v)
             elif k == b"obsolete-hash":
                 obsolete_hash = bin(v)
+            elif k == b"topo-mode":
+                if v == b"pure":
+                    has_pure_topo_heads = True
+                else:
+                    msg = b"unknown topo-mode: %r" % v
+                    raise ValueError(msg)
             else:
                 msg = b"unknown cache key: %r" % k
                 raise ValueError(msg)
         args["key_hashes"] = (filtered_hash, obsolete_hash)
+        if has_pure_topo_heads:
+            pure_line = next(lineiter).rstrip(b'\n')
+            args["pure_topo_branch"] = encoding.tolocal(pure_line)
         return args
 
     def _load_heads(self, repo, lineiter):
         """fully loads the branchcache by reading from the file using the line
         iterator passed"""
         super()._load_heads(repo, lineiter)
+        if self._pure_topo_branch is not None:
+            # no need to read the repository heads, we know their value already.
+            return
         cl = repo.changelog
         getbranchinfo = repo.revbranchcache().branchinfo
         obsrevs = obsolete.getrevs(repo, b'obsolete')
@@ -960,6 +999,62 @@ class BranchCacheV3(_LocalBranchCache):
             self.tiprev,
         )
 
+    def _process_new(
+        self,
+        repo,
+        newbranches,
+        new_closed,
+        obs_ignored,
+        max_rev,
+    ) -> None:
+        if (
+            # note: the check about `obs_ignored` is too strict as the
+            # obsolete revision could be non-topological, but lets keep
+            # things simple for now
+            #
+            # The same apply to `new_closed` if the closed changeset are
+            # not a head, we don't care that it is closed, but lets keep
+            # things simple here too.
+            not (obs_ignored or new_closed)
+            and (
+                not newbranches
+                or (
+                    len(newbranches) == 1
+                    and (
+                        self.tiprev == nullrev
+                        or self._pure_topo_branch in newbranches
+                    )
+                )
+            )
+        ):
+            if newbranches:
+                assert len(newbranches) == 1
+                self._pure_topo_branch = list(newbranches.keys())[0]
+                self._needs_populate = True
+                self._entries.pop(self._pure_topo_branch, None)
+            return
+
+        self._ensure_populated(repo)
+        self._pure_topo_branch = None
+        super()._process_new(
+            repo,
+            newbranches,
+            new_closed,
+            obs_ignored,
+            max_rev,
+        )
+
+    def _ensure_populated(self, repo):
+        """make sure any lazily loaded values are fully populated"""
+        if self._needs_populate:
+            assert self._pure_topo_branch is not None
+            cl = repo.changelog
+            to_node = cl.node
+            topo_heads = self._get_topo_heads(repo)
+            heads = [to_node(r) for r in topo_heads]
+            self._entries[self._pure_topo_branch] = heads
+            self._needs_populate = False
+
 
 class remotebranchcache(_BaseBranchCache):
     """Branchmap info for a remote connection, should not write locally"""
diff --git a/tests/test-branches-obsolete.t b/tests/test-branches-obsolete.t
--- a/tests/test-branches-obsolete.t
+++ b/tests/test-branches-obsolete.t
@@ -113,6 +113,19 @@ We want some branching and some obsolesc
   |
   o  root
   
+
+#if v2
+  $ show_cache
+  ##### .hg/cache/branch2-served
+  3d808bbc94408ea19da905596d4079357a1f28be 8
+  63ba7cd843d1e95aac1a24435befeb1909c53619 o default
+  3d808bbc94408ea19da905596d4079357a1f28be o default
+#else
+  $ show_cache
+  ##### .hg/cache/branch3-served
+  tip-node=3d808bbc94408ea19da905596d4079357a1f28be tip-rev=8 topo-mode=pure
+  default
+#endif
   $ hg log -T '{desc}\n' --rev 'head()'
   B_4
   A_4
@@ -157,7 +170,8 @@ Absolete a couple of changes
 #else
   $ show_cache
   ##### .hg/cache/branch3-served
-  filtered-hash=f8006d64a10d35c011a5c5fa88be1e25c5929514 tip-node=7c29ff2453bf38c75ee8982935739103c38a9284 tip-rev=7
+  filtered-hash=f8006d64a10d35c011a5c5fa88be1e25c5929514 tip-node=7c29ff2453bf38c75ee8982935739103c38a9284 tip-rev=7 topo-mode=pure
+  default
 #endif
   $ cd ..
 
@@ -209,7 +223,6 @@ Revealing tipmost changeset
   $ show_cache
   ##### .hg/cache/branch3
   obsolete-hash=b6d2b1f5b70f09c25c835edcae69be35f681605c tip-node=3d808bbc94408ea19da905596d4079357a1f28be tip-rev=8
-  550bb31f072912453ccbb503de1d554616911e88 o default
   7c29ff2453bf38c75ee8982935739103c38a9284 o default
   ##### .hg/cache/branch3-served
   filtered-hash=f8006d64a10d35c011a5c5fa88be1e25c5929514 obsolete-hash=ac5282439f301518f362f37547fcd52bcc670373 tip-node=3d808bbc94408ea19da905596d4079357a1f28be tip-rev=8
@@ -251,7 +264,8 @@ And we can get back to normal
 #else
   $ show_cache
   ##### .hg/cache/branch3-served
-  filtered-hash=f8006d64a10d35c011a5c5fa88be1e25c5929514 tip-node=7c29ff2453bf38c75ee8982935739103c38a9284 tip-rev=7
+  filtered-hash=f8006d64a10d35c011a5c5fa88be1e25c5929514 tip-node=7c29ff2453bf38c75ee8982935739103c38a9284 tip-rev=7 topo-mode=pure
+  default
 #endif
 
   $ cd ..
@@ -299,7 +313,6 @@ Check that revealing an obsolete changes
   $ show_cache
   ##### .hg/cache/branch3
   obsolete-hash=b6d2b1f5b70f09c25c835edcae69be35f681605c tip-node=3d808bbc94408ea19da905596d4079357a1f28be tip-rev=8
-  550bb31f072912453ccbb503de1d554616911e88 o default
   7c29ff2453bf38c75ee8982935739103c38a9284 o default
   ##### .hg/cache/branch3-served
   filtered-hash=f1456c0d675980582dda9b8edc7f13f503ce544f obsolete-hash=3e74f5349008671629e39d13d7e00d9ba94c74f7 tip-node=7c29ff2453bf38c75ee8982935739103c38a9284 tip-rev=7
@@ -341,7 +354,8 @@ And we can get back to normal
 #else
   $ show_cache
   ##### .hg/cache/branch3-served
-  filtered-hash=f8006d64a10d35c011a5c5fa88be1e25c5929514 tip-node=7c29ff2453bf38c75ee8982935739103c38a9284 tip-rev=7
+  filtered-hash=f8006d64a10d35c011a5c5fa88be1e25c5929514 tip-node=7c29ff2453bf38c75ee8982935739103c38a9284 tip-rev=7 topo-mode=pure
+  default
 #endif
 
   $ cd ..
@@ -434,7 +448,8 @@ And we can get back to normal
 #else
   $ show_cache
   ##### .hg/cache/branch3-served
-  tip-node=7c29ff2453bf38c75ee8982935739103c38a9284 tip-rev=7
+  tip-node=7c29ff2453bf38c75ee8982935739103c38a9284 tip-rev=7 topo-mode=pure
+  default
 #endif
 
   $ cd ..
@@ -477,7 +492,8 @@ Getting the obsolescence marker after th
 #else
   $ show_cache
   ##### .hg/cache/branch3-served
-  tip-node=3d808bbc94408ea19da905596d4079357a1f28be tip-rev=8
+  tip-node=3d808bbc94408ea19da905596d4079357a1f28be tip-rev=8 topo-mode=pure
+  default
 #endif
 
   $ hg pull --rev `cat ../main-single-branch-node_B4` --remote-hidden
@@ -539,7 +555,8 @@ And we can get back to normal
 #else
   $ show_cache
   ##### .hg/cache/branch3-served
-  filtered-hash=f8006d64a10d35c011a5c5fa88be1e25c5929514 tip-node=3d808bbc94408ea19da905596d4079357a1f28be tip-rev=8
+  filtered-hash=f8006d64a10d35c011a5c5fa88be1e25c5929514 tip-node=3d808bbc94408ea19da905596d4079357a1f28be tip-rev=8 topo-mode=pure
+  default
 #endif
 
   $ cd ..
diff --git a/tests/test-branches.t b/tests/test-branches.t
--- a/tests/test-branches.t
+++ b/tests/test-branches.t
@@ -1312,7 +1312,8 @@ Unbundling revision should warm the serv
   0 files updated, 0 files merged, 0 files removed, 0 files unresolved
 #if v3
   $ cat branchmap-update-01/.hg/cache/branch3-base
-  tip-node=99ba08759bc7f6fdbe5304e83d0387f35c082479 tip-rev=1
+  tip-node=99ba08759bc7f6fdbe5304e83d0387f35c082479 tip-rev=1 topo-mode=pure
+  A
 #else
   $ cat branchmap-update-01/.hg/cache/branch2-base
   99ba08759bc7f6fdbe5304e83d0387f35c082479 1
@@ -1327,7 +1328,8 @@ Unbundling revision should warm the serv
   (run 'hg update' to get a working copy)
 #if v3
   $ cat branchmap-update-01/.hg/cache/branch3-served
-  tip-node=71ca9a6d524ed3c2a215119b2086ac3b8c4c8286 tip-rev=3
+  tip-node=71ca9a6d524ed3c2a215119b2086ac3b8c4c8286 tip-rev=3 topo-mode=pure
+  A
 #else
   $ cat branchmap-update-01/.hg/cache/branch2-served
   71ca9a6d524ed3c2a215119b2086ac3b8c4c8286 3
@@ -1356,7 +1358,8 @@ aborted Unbundle should not update the o
 
 #if v3
   $ cat branchmap-update-02/.hg/cache/branch3-base
-  tip-node=99ba08759bc7f6fdbe5304e83d0387f35c082479 tip-rev=1
+  tip-node=99ba08759bc7f6fdbe5304e83d0387f35c082479 tip-rev=1 topo-mode=pure
+  A
 #else
   $ cat branchmap-update-02/.hg/cache/branch2-base
   99ba08759bc7f6fdbe5304e83d0387f35c082479 1
@@ -1372,7 +1375,8 @@ aborted Unbundle should not update the o
   [40]
 #if v3
   $ cat branchmap-update-02/.hg/cache/branch3-base
-  tip-node=99ba08759bc7f6fdbe5304e83d0387f35c082479 tip-rev=1
+  tip-node=99ba08759bc7f6fdbe5304e83d0387f35c082479 tip-rev=1 topo-mode=pure
+  A
 #else
   $ cat branchmap-update-02/.hg/cache/branch2-base
   99ba08759bc7f6fdbe5304e83d0387f35c082479 1