  Query-to-communication lifting theorems, which connect the query complexity
of a Boolean function to the communication complexity of an associated `lifted'
function obtained by composing the function with many copies of another
function known as a gadget, have been instrumental in resolving many open
questions in computational complexity. Several important complexity questions
could be resolved if we could make substantial improvements in the input size
required for lifting with the Index function, from its current near-linear size
down to polylogarithmic in the number of inputs $N$ of the original function
or, ideally, constant. The near-linear size bound was shown by Lovett, Meka,
Mertz, Pitassi and Zhang using a recent breakthrough improvement on the
Sunflower Lemma to show that a certain graph associated with the Index function
of near-linear size is a disperser. They also stated a conjecture about the
Index function that is essential for further improvements in the size required
for lifting with Index using current techniques. In this paper we prove the
following;
  1) The conjecture of Lovett et al. is false when the size of the Index gadget
is $\log N-\omega(1)$.
  2) Also, the Inner-Product function, which satisfies the disperser property
at size $O(\log N)$, does not have this property when its size is $\log
N-\omega(1)$.
  3) Nonetheless, using Index gadgets of size at least 4, we prove a lifting
theorem for a restricted class of communication protocols in which one of the
players is limited to sending parities of its inputs.
  4) Using the ideas from this lifting theorem, we derive a strong lifting
theorem from decision tree size to parity decision tree size. We use this to
derive a general lifting theorem in proof complexity from tree-resolution size
to tree-like $Res(\oplus)$ refutation size, which yields many new exponential
lower bounds on such proofs.
