From the simple reading of the Torah (see Vayikra 5:2-3) it would seem that coming into physical contact with a dead creature that gives off ritual defilement is, itself prohibited. Nevertheless, the tradition of the Sages is that there is nothing inherently wrong with touching such a creature; the only prohibition is for someone who is ritually defiled through such contact to enter the Temple precincts or spread that tumah to something consecrated.
The first Mishnah repeats the teaching that appeared at the beginning of Masechet Shevu’ot (see 2a), that the laws of yediot ha-tumah – recognizing that someone was ritually defiled and then interacted with the Temple or some consecrated object – have the same “two that are four” pattern that parallel the case of shevu’ot – oaths – in that they contain two basic concepts that include four ideas. The two concepts that are written in the Torah are that –
1. he was aware that he was tameh, but then forgot, and touched or ate consecrated food
2. he was aware that he was tameh, but then forgot, and entered the Temple
The Sages then added two further laws –
3. he was aware that he was tameh but forgot that this food is consecrated
4. he was aware that he was tameh but forgot that this is the Temple.
In all four of these cases, he would be obligated to bring a korban oleh ve-yored – a “sliding scale” sacrifice where a wealthy person will bring goat or a lamb, a middle income person will bring a dove and a poor person will bring a meal offering.