If the bottom part of the command module (CM) of the Saturn-V was the thermal shield, how was the CM connected with the utilities of the Service Module (SM)?
They were connected by an umbilical, that curved around the edge of the heat shield.
From CSM06 Command Module Overview page 49 and 50:
For most of an Apollo mission, the command and service modules are attached; they separate only a short time before the command module enters the atmosphere.
The two modules are connected by three tension ties which extend from the CM's aft heat shield to six compression pads on the top of the SM. The tension ties are essentially stainless steel straps about 2-1/2 inches wide and 4 inches long bolted at one end to the CM and at the other to the SM. The CM rests base down on the six compression pads, which are circular metal "cups," three of them about 4 inches in diameter and three about 6 inches in diameter. The areas in the heat shield which rest on the pads are reinforced with laminated fiberglass.
The two modules also are connected through the CM-SM umbilical, an enclosure protruding from the CM on the side opposite the side hatch. The umbilical is the wiring and tubing through which vital power, water, oxygen, and water-glycol flows from one module to the other. These connections are covered by an aluminum fairing about 18 inches wide and 40 inches long.
At separation, electrical circuits are deadfaced (power cut off) and valves closed at the umbilical, a guillotine mechanism cuts the connecting wires and tubing, and small charges sever the tension ties. The umbilical firing pulls away from the CM and remains attached to the SM. The guillotine that severs the wires and tubes consists of two stainless steel blades, either one of which will cut all the connections. The guillotine is driven by redundant detonating cord charges. The tension ties are severed by linear-shaped charges set off by detonators. The signals that set off the detonators, the detonators themselves, and the charges are all redundant.
The area between the bottom (aft) of the CM and the top of the SM where the two modules are joined is enclosed by a fairing 26 inches high. This fairing is part of the service module and contains space radiators for the electrical power subsystem.