It is when the policemen come around the house to search for any trace of the narrator's wife that the narrator's arrogance and total and utter belief in his own ability to conceal the crime reveals itself, and ironically leads to his downfall. Note what the narrator says about his feelings after he has shown the policemen around his house without any fear whatsoever:
The glee at my heart was too strong to be restrained. I burned to say if but one word, by way of triumph, and to render doubly sure their assurance of my guiltlessness.
This feeling of "glee," and desire to show his "triumph" leads him to go to the wall which he built up, concealing his wife's corpse, and to tap on it, stirring the cat that he had inadvertently concealed behind it, and causing it to make a loud noise, therefore attracting the attention of the policemen and leading them to his wife's body, frustrating his attempt to commit the perfect murder. It is the narrator's arrogance therefore that leads to his conviction, as if he had not sought to tap on the wall where his wife's body lay concealed, he would have indeed enacted the perfect murder.