  {\it If gravity is a metric field by Einstein, it is a Higgs field.}
Gravitation theory meets spontaneous symmetry breaking in accordance with the
Equivalence Principle reformulated in the spirit of Klein-Chern geometries of
invariants. In gravitation theory, the structure group of the principal linear
frame bundle $LX$ over a world manifold $X^4$ is reducible to the connected
Lorentz group $SO(3,1)$. The physical underlying reason of this reduction is
Dirac fermion matter possessing only exact Lorentz symmetries. The associated
Higgs field is a tetrad gravitational field $h$ represented by a global section
of the quotient $\Si$ of $LX$ by $SO(3,1)$. The feature of gravity as a Higgs
field issues from the fact that, in the presence of different tetrad fields,
Dirac fermion fields are described by spinor bundles associated with different
reduced Lorentz subbundles of $LX$, and we have nonequivalent representations
of cotangent vectors to $X^4$ by Dirac's matrices. It follows that a fermion
field must be regarded only in a pair with a certain tetrad field. These pairs
fail to be represented by sections of any product bundle $S\times\Si$, but
sections of the composite spinor bundle $S\to\Si\to X^4$. They constitute the
so-called fermion-gravitation complex where values of tetrad gravitational
fields play the role of coordinate parameters, besides the familiar world
coordinates. In Part 1 of the article, geometry of the fermion-gravitation
complex is investigated. The goal is the total Dirac operator into which
components of a connection on $S\to\Si$ along tetrad coordinate directions make
contribution. The Part II will be devoted to dynamics of fermion-gravitation
complex. It is a constraint system to describe which we use the covariant
multisymplectic generalization of the Hamiltonian formalism when canonical
momenta correspond to derivatives of fields with respect to all world
coordinates, not only the time.

General Relativity and Quantum Cosmology