published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

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

Thesis (doctoral)--Cambridge, 1969. / Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1965. / Typescript. Vita. Includes bibliographical references.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1992. / Includes bibliographical references (p. 95-96).

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

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The Periodicity theorem of Hopkins and Smith tells us that any finite spectrum supports a $v_n$-map for some $n$. We are interested in finding finite $2$-local spectra that both support a $v_2$-map with a low power of $v_2$ and have few cells. Following the process outlined in Palmieri-Sadofsky, we study a related class of self-maps, known as $u_2$-maps, between stably finite spectra. We construct examples of spectra that might be expected to support $u_2^1$-maps, and then we use Margolis homology and homological algebra computations to show that they do not support $u_2^1$-maps. We also show that one example does not support a $u_2^2$-map. The nonexistence of $u_2$-maps on these spectra eliminates certain examples from consideration by this technique.

acase@tulane.edu / We study the problem of computing the minimum homotopy area of a planar normal curve. The area of a homotopy is the area swept by the homotopy on the plane. First, we consider a specific class of curves, namely self-overlapping curves, and show that the minimum homotopy area of a self-overlapping curve is equal to its winding area. For an arbitrary normal curve, we show that there is a decomposition of the curve into the self-overlapping subcurves such that the minimum homotopy area can be computed as the sum of winding areas of each self-overlapping subcurve in the decomposition. / 1 / Karakoc, Selcuk