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 H_ {1} (X; \ mathbb {Z}) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4890b3256efb5b4d4f42c6971131eb59bb083cdd)
![X ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18e0e7c566b554eafc1b5705551ac4e939074777)
![{\ displaystyle j_ {n}: H_ {n} (X; \ mathbb {Z}) \ rightarrow H_ {n} (X ^ {+}; \ mathbb {Z})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed42c9d4710b9e484ec61b37c42b4adc37eb74bd)
for all isomorphisms.
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
![\ pi _ {1} X](https://wikimedia.org/api/rest_v1/media/math/render/svg/b73939ad53b202c43c810c0d625ca2d2c10946b8)
![{\ displaystyle \ left \ {D_ {i} \ right \} _ {i \ in I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a11ba35b385877ea88941cb67f17829dde3287b)
![{\ displaystyle e_ {i}: \ partial D_ {i} \ rightarrow X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c572cde7cf43115a0cf6be338614f2f2f99c9ba2)
![X ^ \ prime](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a3a5819cc45f097de14b3ac5a8bedd902bc66d)
![{\ displaystyle 0 \ rightarrow H_ {2} (X) \ rightarrow H_ {2} (X ^ {\ prime}) \ rightarrow H_ {2} (X ^ {\ prime}, X) \ rightarrow 0 = H_ {1 } (X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bc1f0279ca33772d50a100f98cd9454ade223d)
splits because the 2 cells generate free, so one has an isomorphism
![{\ displaystyle H_ {2} (X ^ {\ prime}, X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9c2071cecf4cf67a45528bf8a0bef9e2f972ec)
![{\ displaystyle \ left \ {D_ {i} \ right \} _ {i \ in I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a11ba35b385877ea88941cb67f17829dde3287b)
![{\ displaystyle H_ {2} (X ^ {\ prime}) = H_ {2} (X) \ oplus H_ {2} (X ^ {\ prime}, X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aea87c8e8b5c649a67894999ecc634cb513fc747)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9c2071cecf4cf67a45528bf8a0bef9e2f972ec)
![{\ displaystyle \ left \ {D_ {i} \ right \} _ {i \ in I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a11ba35b385877ea88941cb67f17829dde3287b)
![X ^ \ prime](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a3a5819cc45f097de14b3ac5a8bedd902bc66d)
![{\ displaystyle \ left [D_ {i} \ right] \ in H_ {2} (X ^ {\ prime})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1638f0df5a33ad53dd513c2cb8e45cafdb7f092)
![{\ displaystyle (f_ {i}) _ {*} \ left [S ^ {2} \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95636a147f2d529939a10305c415abd5f4685028)
![{\ displaystyle f_ {i}: S ^ {2} \ rightarrow X ^ {\ prime}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fefc62c90b279921a7c75f9aee92349c60470291)
![{\ displaystyle \ left [S ^ {2} \ right] \ in H_ {2} (S ^ {2}; \ mathbb {Z})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6be385aa71c2dad65598fb561dffd5d4ace6e066)
![{\ displaystyle \ left \ {E_ {i} \ right \} _ {i \ in I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1569677563aec7d51bf41c727eac7b1afe10aadc)
![{\ displaystyle f_ {i}: \ partial E_ {i} \ rightarrow X ^ {\ prime}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dccfe2f394801b59760dcae6c3777c43feb672a7)
![X ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18e0e7c566b554eafc1b5705551ac4e939074777)
![{\ displaystyle H_ {2} (X ^ {+}) = H_ {2} (X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0167e8a4987ebaf33a4a73137dc213f1a37ea1)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ displaystyle H_ {3} (X ^ {+}, X) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c2efff739c8311e6e0c5dcb8a9476ab4cbe265)
![{\ displaystyle H _ {*} (X ^ {+}, X) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e7540e86e7bbf65f8e83c8349f81220590b3d4)
![{\ displaystyle * \ geq 4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c52b470cb325b5ad3ba084febe13be274be8843b)
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
![X ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18e0e7c566b554eafc1b5705551ac4e939074777)
![{\ displaystyle j _ {*}: \ pi _ {1} X \ rightarrow \ pi _ {1} X ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c73dd1829db34eadde94f99076f1f816d46d47fb)
the quotient mapping and the induced morphisms of the homology groups
![{\ displaystyle j_ {n}: H_ {n} (X; \ mathbb {Z}) \ rightarrow H_ {n} (X ^ {+}; \ mathbb {Z})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed42c9d4710b9e484ec61b37c42b4adc37eb74bd)
for all isomorphisms.
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5ef659b291303fd08bd4da1d9e17f637e12d10)
![N](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3)
![{\ displaystyle \ left \ {D_ {i} \ right \} _ {i \ in I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a11ba35b385877ea88941cb67f17829dde3287b)
![{\ displaystyle e_ {i}: \ partial D_ {i} \ rightarrow X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c572cde7cf43115a0cf6be338614f2f2f99c9ba2)
![X ^ \ prime](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a3a5819cc45f097de14b3ac5a8bedd902bc66d)
![{\ displaystyle X \ to X ^ {\ prime}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79fdb6335b3f33ddb2da53996ca1abbff75794c0)
![{\ displaystyle \ pi _ {1} X \ to \ pi _ {1} X / N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba94a7ea5ea2c35fe46a2da48ad668c661c56886)
![{\ displaystyle {\ widetilde {X ^ {\ prime}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d02dcbe66f11b6a4dd25576c1edd26ea678c807)
![X ^ \ prime](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a3a5819cc45f097de14b3ac5a8bedd902bc66d)
![{\ displaystyle {\ widetilde {X}} \ subset {\ widetilde {X ^ {\ prime}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5748cd31f09cececd0ad23962236943a29c790)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ displaystyle \ pi _ {1} {\ widetilde {X}} = N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4e4571c95df60df3d1ce4906d7b924e2f1752d)
![N](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3)
![{\ displaystyle H_ {1} ({\ widetilde {X}}) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edab365c2fc3c4e4ce6d08639efec82a0fd95b0a)
![{\ displaystyle H_ {2} ({\ widetilde {X ^ {\ prime}}}) = H_ {2} ({\ widetilde {X}}) \ oplus H_ {2} ({\ widetilde {X ^ {\ prime}}}, {\ widetilde {X}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc86a046e4a4c620e63de7a38831fd8edeb3b33)
![{\ displaystyle H_ {2} ({\ widetilde {X ^ {\ prime}}}, {\ widetilde {X}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b95ae9b698b6e7b34e36023ba1015835e2125e3)
![{\ displaystyle \ left \ {D_ {i} \ right \} _ {i \ in I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a11ba35b385877ea88941cb67f17829dde3287b)
![{\ displaystyle \ mathbb {Z} \ left [\ pi _ {1} X / N \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edc1d5cd9c7197a7f4c7aa9a0ab09d14f5fe9583)
![{\ displaystyle {\ widetilde {X ^ {\ prime}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d02dcbe66f11b6a4dd25576c1edd26ea678c807)
![{\ displaystyle \ left [D_ {i} \ right] \ in H_ {2} (X ^ {\ prime})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1638f0df5a33ad53dd513c2cb8e45cafdb7f092)
![{\ displaystyle f_ {i}: S ^ {2} \ rightarrow X ^ {\ prime}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fefc62c90b279921a7c75f9aee92349c60470291)
![{\ displaystyle \ left \ {E_ {i} \ right \} _ {i \ in I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1569677563aec7d51bf41c727eac7b1afe10aadc)
![{\ displaystyle f_ {i}: \ partial E_ {i} \ rightarrow X ^ {\ prime}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dccfe2f394801b59760dcae6c3777c43feb672a7)
![X ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18e0e7c566b554eafc1b5705551ac4e939074777)
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 .
![f \ colon X \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e)
![{\ displaystyle N_ {X} \ subset \ pi _ {1} X, N_ {Y} \ subset \ pi _ {1} Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56e8a5806a2c0e66f2fd4800181e7a82b9f66b66)
![{\ displaystyle f _ {*} (N_ {X}) \ subset N_ {Y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7edfdf30da6be36caf667a6fe9e2cccd3c0e06b)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![{\ displaystyle f ^ {+} \ colon X ^ {+} \ to Y ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e8414608d3f50ab8b6bedf40cca44747dc2bcb)
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 .
![BG](https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![N \ subset G](https://wikimedia.org/api/rest_v1/media/math/render/svg/66124fee39820b9a045a07c657b752eb84b1e869)
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
![{\ displaystyle BG \ to BG ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/149d3f913136c374f32d2e155ad42ed529f23b8a)
![\ pi _ {1} F](https://wikimedia.org/api/rest_v1/media/math/render/svg/79c80a5865e1eae93d5de4e5fe14580965b0f84e)
![N](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3)
![{\ displaystyle \ pi _ {2} (BG ^ {+}) = H_ {2} (N; \ mathbb {Z})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67524a7b721bc692cbbab5d19090b345d6407ab9)
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
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![{\ displaystyle GL (R) = \ bigcup _ {n \ geq 0} GL (n, R)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e6080caba60e05b092b191467d5f7825481e8a5)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![{\ displaystyle BGL (R)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14cd86282c5d34dd0343f43226a149d61dbebdf9)
![{\ displaystyle GL (R)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a50a90fda013ad80140f9e48d303d27fb876857)
![{\ displaystyle GL (R)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a50a90fda013ad80140f9e48d303d27fb876857)
![{\ displaystyle E (R) = \ left [GL (R), GL (R) \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4699ff13af6dc653f8c431db62a14eb7ebb2a0fb)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![{\ displaystyle K_ {i} (R): = \ pi _ {i} (BGL ^ {+} (R))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e12cd384d988005c6db2a05d8f07d559a1816764)
for .
![i \ ge 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/a40d31039a220c00dcc836babe2c5f6961c689bf)
Example: finite bodies
Let be a finite field with elements, then according to a Quillen theorem there is a homotopy equivalence![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
-
,
being the fiber of the figure
![{\ displaystyle E \ Psi ^ {q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78825c36acb4758f6c12b0c56c13a284759747c1)
![{\ displaystyle \ Psi ^ {q} -Id: BU \ rightarrow BU}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62143c03a42ac7cd6d58ba89672ac7e544bd53f7)
(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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4032e58bd7012b887eeea151a90322b58c496670)
![{\ displaystyle E \ Psi ^ {q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78825c36acb4758f6c12b0c56c13a284759747c1)
-
.
H-space
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d06fb351cbca6ab55b8dcddcd40df19f0bc6db)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![BGL ^ {+} (R)](https://wikimedia.org/api/rest_v1/media/math/render/svg/26c4f9c6f88aade61ec09726d660d57f83628789)
literature
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Daniel Quillen : Cohomology of groups. Actes Congrès Internat. Math., 2, Gauthier-Villars (1973) pp. 47-51 pdf
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Jonathan Rosenberg : Algebraic K-theory and its applications. Graduate Texts in Mathematics, 147. Springer-Verlag, New York, 1994. ISBN 0-387-94248-3
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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
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Allen Hatcher : Algebraic topology. Cambridge University Press, Cambridge, 2002. ISBN 0-521-79160-X pdf
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Jean-Claude Hausmann ; Dale Husemoller : Acyclic maps. Enseign. Math. (2) 25 (1979) no. 1-2, 53-75
Web links
Individual evidence
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↑ Rosenberg, op.cit., Proposition 5.2.4
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^ Weibel, op.cit., Proposition IV.1.7
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^ Jean Louis Loday : Structure multiplicative en K-théorie algébrique. CR Acad. Sci. Paris Sér. 1974 A 279: 321-324.