In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups.
is a based connected CW complex and
is a perfect normal subgroup of
π
then a map
is called a +-construction relative to
induces an isomorphism on homology, and
is the kernel of
[1] The plus construction was introduced by Michel Kervaire (1969), and was used by Daniel Quillen to define algebraic K-theory.
Given a perfect normal subgroup of the fundamental group of a connected CW complex
, attach two-cells along loops in
whose images in the fundamental group generate the subgroup.
This operation generally changes the homology of the space, but these changes can be reversed by the addition of three-cells.
The most common application of the plus construction is in algebraic K-theory.
is a unital ring, we denote by
the group of invertible
matrices with elements in
by attaching a
The direct limit of these groups via these maps is denoted
and its classifying space is denoted
The plus construction may then be applied to the perfect normal subgroup
, generated by matrices which only differ from the identity matrix in one off-diagonal entry.
-th homotopy group of the resulting space,