A popular what if on this site is to ask what would have happened if Harold II had won the Battle of Hastings in 1066. I am (reasonably) certain, however, that nobody has ever asked what would have happened had William the Conqueror been infertile. In our timeline, when William took the throne six (possibly seven) of his nine children had been born (though crucially, not the future King Henry I, or Adela, mother of King Stephen). But let's apply handwavium here, and say that he has no children, but that this doesn't affect him taking the throne.
Presumably he keeps trying for a child; his apparent faithfulness to Matilda of Flanders in OTL is probably down to the fact that she provided nine children for him. In this timeline he probably takes a string of mistresses, especially given as he has no reason to be opposed to his being succeeded by a bastard, having been one himself. But these attempts to produce an illegitimate heir will be unsuccessful, given his infertility. Presumably he would also try to remarry in 1083, or whenever Matilda of Flanders dies in this timelime, in the hope that he finds a fertile wife. Any guesses on who he might marry?
After he dies, who would succeed him? His nephew, Stephen, Count of Aumale, is probably the closest male relation descended from William's father, Robert, Duke of Normandy. Otherwise there's always his half brothers, Odo, Earl of Kent, and Robert, Count of Mortain. Odo had acted as regent for William when he was not in the country (presumably he does so in this timeline), but given his being a bishop, he's unlikely to be the one to secure the succession to the throne, though as a powerful landowner he would be in a strong position - though this assumes he avoids jail for defrauding the crown. Robert was apparently somewhat more favoured at the end of William's life, and was also a major landowner. Also in his favour is the fact that he had a son, William. Finally for your consideration, Edgar the Ætheling was still alive and had a claim to the throne, though he's likely to require foreign support, given that the Anglo-Saxon nobility had largely been replaced by the Normans.
Presumably he keeps trying for a child; his apparent faithfulness to Matilda of Flanders in OTL is probably down to the fact that she provided nine children for him. In this timeline he probably takes a string of mistresses, especially given as he has no reason to be opposed to his being succeeded by a bastard, having been one himself. But these attempts to produce an illegitimate heir will be unsuccessful, given his infertility. Presumably he would also try to remarry in 1083, or whenever Matilda of Flanders dies in this timelime, in the hope that he finds a fertile wife. Any guesses on who he might marry?
After he dies, who would succeed him? His nephew, Stephen, Count of Aumale, is probably the closest male relation descended from William's father, Robert, Duke of Normandy. Otherwise there's always his half brothers, Odo, Earl of Kent, and Robert, Count of Mortain. Odo had acted as regent for William when he was not in the country (presumably he does so in this timeline), but given his being a bishop, he's unlikely to be the one to secure the succession to the throne, though as a powerful landowner he would be in a strong position - though this assumes he avoids jail for defrauding the crown. Robert was apparently somewhat more favoured at the end of William's life, and was also a major landowner. Also in his favour is the fact that he had a son, William. Finally for your consideration, Edgar the Ætheling was still alive and had a claim to the throne, though he's likely to require foreign support, given that the Anglo-Saxon nobility had largely been replaced by the Normans.
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