 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 Algebraic homotopy theory, groups, and Ktheory
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
Algebraic homotopy theory, groups, and Ktheory Jardine, J. F.
Abstract
Let Mk be the category of algebras over a unique factorization domain k, and let indAffk denote the category of prorepresentable functors from Mk to the category E of sets. It is shown that indAffk is a closed model category in such a way that its associated homotopy category Ho(indAffk) is equivalent to the homotopy category Ho(S) which comes from the category S of simplicial sets. The equivalence is induced by functors Sk: indAffk > S and Rk: S> indAffk. In an effort to determine what is measured by the homotopy groups πi(X) := πi. (Sk X) of X in indAffk in the case where k is an algebraically closed field, some homotopy groups of affine reduced algebraic groups G over k are computed. It is shown that, if G is connected, then π₀ (G) = * if and only if the group G(k) of krational points of G is generated by unipotents. A fibration theory is developed for homomorphisms of algebraic groups which are surjective on rational points which allows the computation of the homotopy groups of any connected algebraic group G in terms of the homotopy groups of the universal covering groups of the simple algebraic subgroups of the associated semisimple group G/R(G), where R(G) is the solvable radical of G. The homotopy groups of simple Chevalley groups over almost all fields k are studied. It is shown that the homotopy groups of the special linear groups S1n and of the symplectic groups Sp2m converge, respectively, to the Ktheory and ₋₁Ltheory of the underlying field k. It is shown that there are isomorphisms π₁ (S1n ) = H₂(S1n (k);Z) = K₂(k) for n ≥ 3 and almost all fields k, and π₁ (Sp₂m ) = H₂(Sp₂m) (k);Z) = ₋₁L₂(k) for m ≥ 1 and almost all fields k of characteristic ≠ 2, where Z denotes the ring of integers. It is also shown that π₁(Sp₂m) = H₂(Sp2m(k);Z) = K₂ (k) if k is algebraically closed of arbitrary characteristic. A spectral sequence for the homology of the classifying space of a simplicial group is used for all of these calculations.
Item Metadata
Title 
Algebraic homotopy theory, groups, and Ktheory

Creator  
Publisher 
University of British Columbia

Date Issued 
1981

Description 
Let Mk be the category of algebras over a unique factorization
domain k, and let indAffk denote the category of prorepresentable functors from Mk to the category E of sets. It is shown that
indAffk is a closed model category in such a way that its associated homotopy category Ho(indAffk) is equivalent to the homotopy category Ho(S) which comes from the category S of simplicial sets. The
equivalence is induced by functors Sk: indAffk > S and
Rk: S> indAffk.
In an effort to determine what is measured by the homotopy groups πi(X) := πi. (Sk X) of X in indAffk in the case where k is
an algebraically closed field, some homotopy groups of affine reduced algebraic groups G over k are computed. It is shown that, if G is connected, then π₀ (G) = * if and only if the group G(k) of krational points of G is generated by unipotents. A fibration theory is developed for homomorphisms of algebraic groups which are surjective on rational points which allows the computation of the homotopy groups of any connected algebraic group G in terms of the homotopy groups of the universal covering groups of the simple algebraic subgroups of the associated semisimple group G/R(G), where R(G) is the solvable radical of G.
The homotopy groups of simple Chevalley groups over almost all
fields k are studied. It is shown that the homotopy groups of the
special linear groups S1n and of the symplectic groups Sp2m converge,
respectively, to the Ktheory and ₋₁Ltheory of the underlying field k. It is shown that there are isomorphisms
π₁ (S1n ) = H₂(S1n (k);Z) = K₂(k) for n ≥ 3 and almost all fields k, and π₁ (Sp₂m ) = H₂(Sp₂m) (k);Z) = ₋₁L₂(k) for m ≥ 1 and almost all fields k of characteristic ≠ 2, where Z denotes the ring of integers. It is also shown that π₁(Sp₂m) = H₂(Sp2m(k);Z) = K₂ (k) if k is algebraically closed of arbitrary characteristic. A spectral sequence for the homology of the classifying space of a simplicial group is used for all of these calculations.

Genre  
Type  
Language 
eng

Date Available 
20100330

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080143

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.