Thesis (Ph. D.)--Lehigh University, 2004. / Includes vita. Includes bibliographical references (leaves 93-94).

Thesis (Ph.D.)--Aberdeen University, 2009. / Title from web page (viewed on July 20, 2009). Includes bibliographical references.

No description available.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2003. / Includes bibliographical references. Also available on the Internet.

Let Mk be the category of algebras over a unique factorization domain k, and let ind-Affk denote the category of pro-representable functors from Mk to the category E of sets. It is shown that ind-Affk is a closed model category in such a way that its associated homotopy category Ho(ind-Affk) is equivalent to the homotopy category Ho(S) which comes from the category S of simplicial sets. The equivalence is induced by functors Sk: ind-Affk -> S and Rk: S-> ind-Affk. In an effort to determine what is measured by the homotopy groups πi(X) := πi. (Sk X) of X in ind-Affk 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 k-rational 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 semi-simple 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 K-theory and ₋₁L-theory 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. / Science, Faculty of / Mathematics, Department of / Graduate

A fibrewise coHopf space X over a base B is a sectioned space for which the diagonal map X —> X x _BX may be compressed into X V_BX up to fibrewise pointed homotopy. Such spaces have been investigated by I. M. James in the case where X is a sphere bundle over a sphere. The purpose of this thesis is to demonstrate some of the properties of fibrewise coHopf spaces over more general bases. Particular attention is given to sphere bundles and fibrations with spherical fibre. The fibrewise reduced suspension of a sectioned fibrewise space with closed sec- tion is fibrewise coHopf with associative comultiplication (up to fibrewise pointed homotopy) and a fibrewise inversion. Examples of fibrewise coHopf spaces not of this form are exhibited, and sufficient conditions are given to ensure that a fibrewise coHopf space has the primitive fibrewise pointed homotopy type of a fibrewise re- duced suspension, in terms of the dimension and connectivity of the space, its base and the fibres. It is shown that these conditions may be relaxed if the fibrewise coHopf structure on the space is assumed to be homotopy-associative. An example of a non-associative fibrewise coHopf sphere bundle is given. It is shown that, if q > 1 is odd, a sectioned orientable q-sphere bundle over a finite connected complex is fibrewise coHopf if and only if its fibrewise localisation at the prime 2 is fibrewise coHopf. Moreover, the fibrewise rationalisation of an odd-dimensional sphere bundle over a finite polyhedron whose fibrewise unreduced suspension is fibrewise coHopf is shown to be a trivial fibration. As an application, it is shown that new fibrewise coHopf spherical fibrations may be constructed by mixing. The Thorn space is used to determine the cohomology ring of the total space of a fibrewise coHopf sphere bundle in terms of that of its base, and a generalised Hopf invariant is constructed which vanishes on fibrewise coHopf sphere bundles.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (leaves 48-49).

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2007. / Includes bibliographical references (p. 65-66).

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2005. / Includes bibliographical references (leaf 15).

Typescript. Includes vita and abstract. Includes bibliographical references (leaves 53-54) Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.