Plus construction

The plus construction (often referred to as Quillens plus construction ) is a method of algebraic topology that is used, among other things, to define the algebraic K-theory .

construction

Construction in the case of perfect fundamental groups

Theorem : Let be a connected CW-complex with . Then there is a simply connected CW complex constructed by gluing 2 and 3 cells together and an inclusion , so that the induced morphisms of the homology groups ${\displaystyle X}$ ${\displaystyle H_{1}(X;\mathbb {Z} )=0}$ ${\displaystyle X^{+}}$ ${\displaystyle j:X\rightarrow X^{+}}$

${\displaystyle j_{n}:H_{n}(X;\mathbb {Z} )\rightarrow H_{n}(X^{+};\mathbb {Z} )}$

for all isomorphisms. ${\displaystyle n\in \mathbb {N} }$

Construction / idea of ​​proof: Be representatives for a generating system of the fundamental group . By attaching 2 cells using the images , a simply connected CW complex is obtained . The long exact sequence ${\displaystyle \left\{e_{i}:S^{1}\rightarrow X\right\}_{i\in I}}$ ${\displaystyle \pi _{1}X}$ ${\displaystyle \left\{D_{i}\right\}_{i\in I}}$ ${\displaystyle e_{i}:\partial D_{i}\rightarrow X}$ ${\displaystyle X^{\prime }}$

${\displaystyle 0\rightarrow H_{2}(X)\rightarrow H_{2}(X^{\prime })\rightarrow H_{2}(X^{\prime },X)\rightarrow 0=H_{1}(X)}$

splits because the 2 cells generate free, so one has an isomorphism ${\displaystyle H_{2}(X^{\prime },X)}$ ${\displaystyle \left\{D_{i}\right\}_{i\in I}}$

${\displaystyle H_{2}(X^{\prime })=H_{2}(X)\oplus H_{2}(X^{\prime },X)}$

and the summand is generated by the . Because simply is connected, according to Hurewicz's theorem, the elements are of the form for illustrations . (Herein , the fundamental class .) By adhering 3 cells by means of the images to obtain a single contiguous CW complex with . Because the adhered three cells their edge not in who applies , and because only 2- and 3-dimensional cells were adhered applies for . So one also has an isomorphism for all homology groups from grade 3 onwards. ${\displaystyle H_{2}(X^{\prime },X)}$ ${\displaystyle \left\{D_{i}\right\}_{i\in I}}$ ${\displaystyle X^{\prime }}$ ${\displaystyle \left[D_{i}\right]\in H_{2}(X^{\prime })}$ ${\displaystyle (f_{i})_{*}\left[S^{2}\right]}$ ${\displaystyle f_{i}:S^{2}\rightarrow X^{\prime }}$ ${\displaystyle \left[S^{2}\right]\in H_{2}(S^{2};\mathbb {Z} )}$ ${\displaystyle \left\{E_{i}\right\}_{i\in I}}$ ${\displaystyle f_{i}:\partial E_{i}\rightarrow X^{\prime }}$ ${\displaystyle X^{+}}$ ${\displaystyle H_{2}(X^{+})=H_{2}(X)}$ ${\displaystyle X}$ ${\displaystyle H_{3}(X^{+},X)=0}$ ${\displaystyle H_{*}(X^{+},X)=0}$ ${\displaystyle *\geq 4}$

Construction in the general case

Theorem : Let be a connected CW-complex and a perfect normal divisor . Then there is a CW complex constructed by gluing 2 and 3 cells and an inclusion so that the induced morphism of the fundamental groups ${\displaystyle X}$ ${\displaystyle N\subset \pi _{1}X}$ ${\displaystyle X^{+}}$ ${\displaystyle j:X\rightarrow X^{+}}$

${\displaystyle j_{*}:\pi _{1}X\rightarrow \pi _{1}X^{+}}$

the quotient mapping and the induced morphisms of the homology groups ${\displaystyle \pi _{1}X\to \pi _{1}X/N}$

${\displaystyle j_{n}:H_{n}(X;\mathbb {Z} )\rightarrow H_{n}(X^{+};\mathbb {Z} )}$

for all isomorphisms. ${\displaystyle n\in \mathbb {N} }$

Construction / idea of ​​proof: Be representatives for a generating system of . By gluing 2 cells using the images , a CW complex is obtained , so that the homomorphism of the fundamental groups generated by the inclusion is the quotient image . Let be the universal superposition of and the archetype of , so and (because is perfect) . Analogous to the above one has an isomorphism and the summand is the free module generated by the . Because it is simply connected, there are realizing images and by gluing 3 cells by means of the images one again obtains a simply connected CW complex with the desired properties. ${\displaystyle \left\{e_{i}:S^{1}\rightarrow X\right\}_{i\in I}}$ ${\displaystyle N}$ ${\displaystyle \left\{D_{i}\right\}_{i\in I}}$ ${\displaystyle e_{i}:\partial D_{i}\rightarrow X}$ ${\displaystyle X^{\prime }}$ ${\displaystyle X\to X^{\prime }}$ ${\displaystyle \pi _{1}X\to \pi _{1}X/N}$ ${\displaystyle {\widetilde {X^{\prime }}}}$ ${\displaystyle X^{\prime }}$ ${\displaystyle {\widetilde {X}}\subset {\widetilde {X^{\prime }}}}$ ${\displaystyle X}$ ${\displaystyle \pi _{1}{\widetilde {X}}=N}$ ${\displaystyle N}$ ${\displaystyle H_{1}({\widetilde {X}})=0}$ ${\displaystyle H_{2}({\widetilde {X^{\prime }}})=H_{2}({\widetilde {X}})\oplus H_{2}({\widetilde {X^{\prime }}},{\widetilde {X}})}$ ${\displaystyle H_{2}({\widetilde {X^{\prime }}},{\widetilde {X}})}$ ${\displaystyle \left\{D_{i}\right\}_{i\in I}}$ ${\displaystyle \mathbb {Z} \left[\pi _{1}X/N\right]}$ ${\displaystyle {\widetilde {X^{\prime }}}}$ ${\displaystyle \left[D_{i}\right]\in H_{2}(X^{\prime })}$ ${\displaystyle f_{i}:S^{2}\rightarrow X^{\prime }}$ ${\displaystyle \left\{E_{i}\right\}_{i\in I}}$ ${\displaystyle f_{i}:\partial E_{i}\rightarrow X^{\prime }}$ ${\displaystyle X^{+}}$

Functoriality

Let it be a continuous mapping between connected CW-complexes and let it be perfect normal divisors with . Then induces an unambiguous continuous continuation except for homotopy . ${\displaystyle f\colon X\to Y}$ ${\displaystyle N_{X}\subset \pi _{1}X,N_{Y}\subset \pi _{1}Y}$ ${\displaystyle f_{*}(N_{X})\subset N_{Y}}$ ${\displaystyle f}$ ${\displaystyle f^{+}\colon X^{+}\to Y^{+}}$

Homotopy fiber

Be the classifying space of a discrete group and a perfect normal divider. If the homotopy fiber is the plus construction , then the universal central extension of and . ${\displaystyle BG}$ ${\displaystyle G}$ ${\displaystyle N\subset G}$ ${\displaystyle F}$ ${\displaystyle BG\to BG^{+}}$ ${\displaystyle \pi _{1}F}$ ${\displaystyle N}$ ${\displaystyle \pi _{2}(BG^{+})=H_{2}(N;\mathbb {Z} )}$

Algebraic K theory

→ Main article : Algebraic K-theory

Let be a unitary ring , the group of invertible matrices above and the classifying space of , i.e. H. an aspherical space with a fundamental group . Because the group of elementary matrices is perfect and a normal divisor, the plus construction can be used. The algebraic K-theory of the ring is defined as ${\displaystyle R}$ ${\displaystyle GL(R)=\bigcup _{n\geq 0}GL(n,R)}$ ${\displaystyle R}$ ${\displaystyle BGL(R)}$ ${\displaystyle GL(R)}$ ${\displaystyle GL(R)}$ ${\displaystyle E(R)=\left[GL(R),GL(R)\right]}$ ${\displaystyle R}$

${\displaystyle K_{i}(R):=\pi _{i}(BGL^{+}(R))}$

for . ${\displaystyle i\geq 1}$

Example: finite bodies

Let be a finite field with elements, then according to a Quillen theorem there is a homotopy equivalence ${\displaystyle K}$ ${\displaystyle q}$

${\displaystyle BGL(k)^{+}\simeq E\Psi ^{q}}$,

being the fiber of the figure ${\displaystyle E\Psi ^{q}}$

${\displaystyle \Psi ^{q}-Id:BU\rightarrow BU}$

(for the effect of the Adams operation on the classifying space of the unitary group ) is. The homotopy groups of can be calculated with Bott periodicity , the result is ${\displaystyle \Psi ^{q}}$ ${\displaystyle E\Psi ^{q}}$

${\displaystyle K_{2i}(K)=0,K_{2i-1}(K)=\mathbb {Z} /(q^{i}-1)\mathbb {Z} }$.

H-space

${\displaystyle BGL^{+}(R)}$is an H-space using a link defined by Loday. The plus construction is universal for images in H-spaces, i.e. H. every continuous mapping into an H-space is factored over . ${\displaystyle BGL(R)\to H}$ ${\displaystyle H}$ ${\displaystyle BGL^{+}(R)}$

literature

• Daniel Quillen : Cohomology of groups. Actes Congrès Internat. Math., 2, Gauthier-Villars (1973) pp. 47-51 pdf
• Jonathan Rosenberg : Algebraic K-theory and its applications. Graduate Texts in Mathematics, 147. Springer-Verlag, New York, 1994. ISBN 0-387-94248-3
• Charles Weibel : The K-book. An introduction to algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN 978-0-8218-9132-2
• Allen Hatcher : Algebraic topology. Cambridge University Press, Cambridge, 2002. ISBN 0-521-79160-X pdf
• Jean-Claude Hausmann ; Dale Husemoller : Acyclic maps. Enseign. Math. (2) 25 (1979) no. 1-2, 53-75

Web links

• Plus construction (Encyclopedia of Mathematics)
• Shah: The Quillen plus construction in algebraic K-theory
• Hausmann: Homology bordism and Quillen plus construction
• Friedlander: An introduction to K-theory

Individual evidence

1. ↑ Rosenberg, op.cit., Proposition 5.2.4
2. ^ Weibel, op.cit., Proposition IV.1.7
3. ^ Jean Louis Loday : Structure multiplicative en K-théorie algébrique. CR Acad. Sci. Paris Sér. 1974 A 279: 321-324.