  The homotopical information hidden in a supersymmetric structure is revealed
by considering deformations of a configuration manifold. This is in sharp
contrast to the usual standpoints such as Connes' programme where a geometrical
structure is rigidly fixed. For instance, we can relate supersymmetries of
types N=2n and N=(n, n) in spite of their gap due to distinction between
$\Bbb{Z}_2$(even-odd)- and integer-gradings.
  Our approach goes beyond the theory of real homotopy due to Quillen, Sullivan
and Tanr\'e developed, respectively, in the 60's, 70's and 80's, which exhibits
real homotopy of a 1-connected space out of its de Rham-Fock complex with
supersymmetry. Our main new step is based upon the Taylor (super-)expansion and
locality, which links differential geometry with homotopy without the
restriction of 1-connectedness. While the homotopy invariants treated so far in
relation with supersymmetry are those depending only on $\Bbb{Z}_2$-grading
like the index, here we can detect new $\Bbb{N}$-graded homotopy invariants.
While our setup adopted here is (graded) commutative, it can be extended also
to the non-commutative cases in use of state germs (Haag-Ojima) corresponding
to a Taylor expansion.
