A rapier would be out of place, but most other things would be okay. Historically a polearm was really any random bit of iron the local blacksmith thought looked good on the end of a stick, it's really modern museam curators who insist on classifying everything to death. Only items which were used in tourneys (like pole axes) were anything like standardized.
Polearms were easy to make, but they were much more sophisticated than just "put some pointy metal on a stick". Both the halberd and the glaive for example would not exist in the 11th century (being invented in the 14th and 16th century). And they had a very specific form and were not just "junk on wood".
And most heavy weapons were invented to counter heavy armour, but exactly this armour did not exist at that time frame, ergo the weapons would also be missing.
For example maces and war hammers wouldn't be really wide spread at that point (war hammers maybe not even invented) as there was no real reason to use them. And no greatswords, too.
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