In what sense has De Maupassant provided a false climax toward the end of "The Piece of String"?
The climax of the story that you are referring to occurs when the purse and the 500 francs are returned, therefore, Hauchecome believes, as does the reader, that he will be vindicated, now that the truth is known. He did not steal the purse, he was not involved in the theft in any way. But what happens, instead of Hauchecome being viewed as innocent, he is viewed as more treacherous, accused of having an accomplice for the crime.
Hauchecome can find no peace from the wrongful accusation. It does not matter that the real thief returned the purse with the money. The people of the village have labeled him, permanently, as a villain.
All the hope he felt when the purse was returned leads to nothing. It is anti-climatic, as Hauchecome is still tormented by both his internal and external conflicts. Intenally he is haunted with anxiety over the false accusation, muttering on his death bed "it was only a piece of string."
Externally, the people in the village, as well as his arch enemy, Malandain, believe him to be a guilty, manipulative man. He permanently loses his status as an honest man, although known as a shrewd businessman, now the town permanently labels him a thief.