Plus construction explained

In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups.

Explicitly, if

X is a based connected CW complex and P is a perfect normal subgroup of \pi _{1(X)} then a map f\colon X \to Y is called a +-construction relative to P if f induces an isomorphism on homology, and P is the kernel of \pi _{1(X)} \to \pi _{1(Y)} .^[1] X , attach two-cells along loops in X 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. If

R is a unital ring, we denote by \operatorname{GL} _{n(R)} the group of invertible n -by-n matrices with elements in R . \operatorname{GL} _{n(R)} embeds in \operatorname{GL} _{n+1} (R) by attaching a 1 along the diagonal and 0 s elsewhere. The direct limit of these groups via these maps is denoted \operatorname{GL}(R) and its classifying space is denoted B\operatorname{GL}(R) . The plus construction may then be applied to the perfect normal subgroup E(R) of \operatorname{GL}(R) = \pi _{1(B\operatorname{GL}(R))} , generated by matrices which only differ from the identity matrix in one off-diagonal entry. For n> 0 , the n -th homotopy group of the resulting space, B\operatorname{GL}(R) ^{+} , is isomorphic to the n -th K -group of R , that is, \pi _{n\left(} B\operatorname{GL}(R) ^{+\right)} \cong K _{n(R).}

See also

• Semi-s-cobordism

References

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Notes and References

1. [Charles Weibel]