No, the trackers only say the ip / port and a flag (1 or 0) that indicates whether that host is seed or leech (100% or less of the torrent content). To get the rest of the information, it is necessary to connect to one of the machines that the tracker indicates and request the other information using the BitTorrent protocol.
Take a look at this: link
More precisely in defining this message:
bitfield: len = 0001 + X id = 5 bitfield The bitfield message may only be
immediately after the handshaking sequence is completed, and
before any other messages are sent. It is optional, and it need not be
Sent if a client has no pieces.
The bitfield message is variable length, where X is the length of the
bitfield. The payload is a bitfield representing the pieces that have
been successfully downloaded. The high bit in the first byte
Bits that are cleared to
missing piece, and set bits indicate to valid and available piece.
Spare bits at the end are set to zero.
Some clients (Deluge for example) send bitfield with missing pieces
even if it has all data. Then it sends rest of pieces as have
messages. They are saying this helps against ISP filtering of
BitTorrent protocol. It's called lazy bitfield.
A bitfield of the wrong length is considered an error. Clients should
drop the connection if they receive bitfields that are not of the
correct size, or if the bitfield has any of the spare bits set.
I believe that your question has already been answered in the first sentence, but to complement the information, there is a translation (half mouth and well summarized, from the definition of the message I posted above):
After negotiating the connection, both peers should send a type
specific message, before any other, containing a bit sequence, which represents the sequence of chunks of the torrent contents. The bits set with 1 represent the pieces that the peer has, and the bits set with 0, the pieces that he does not have. If the peer does not have any parts, it does not need to send this message.