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This thesis proposes an improved notion of coefficient system for Lagrangian Floer Homology which allows one to produce nontrivial invariants away from characteristic 2, even when coherent orientations of moduli spaces of Floer trajectories do not exist. This explains a suggestion of Witten. The invariant can be computed in examples, and the method explained below should be extensible to other Floer-theoretic invariants. The basic idea is that the moduli spaces of curves admit fundamental classes in homology with coefficients in the orientation lines of the moduli spaces, and the usual construction of coherent orientations actually shows that these fundamental classes naturally map to spaces of paths twisted with appropriate coefficient systems. These twisted path spaces admit enough algebraic structure to make sense of Floer homology with coefficients in these path spaces.

There is an ongoing massive effort by many researchers to link category theory and geometry, especially homotopy coherence and categorical coherence. This constitutes just a part of the broad undertaking known as categorification as described by Baez and Dolan. This effort has as a partial goal that of understanding the categories and functors that correspond to loop spaces and their associated topological functors. Progress towards this goal has been advanced greatly by the recent work of Balteanu, Fiedorowicz, Schwänzl, and Vogt who show a direct correspondence between k–fold monoidal categories and k–fold loop spaces through the categorical nerve. This thesis pursues the hints of a categorical delooping that are suggested when enrichment is iterated. At each stage of successive enrichments, the number of monoidal products seems to decrease and the categorical dimension to increase, both by one. This is mirrored by topology. When we consider the loop space of a topological space, we see that paths (or 1–cells) in the original are now points (or objects) in the derived space. There is also automatically a product structure on the points in the derived space, where multiplication is given by concatenation of loops. Delooping is the inverse functor here, and thus involves shifting objects to the status of 1–cells and decreasing the number of ways to multiply. Enriching over the category of categories enriched over a monoidal category is defined, for the case of symmetric categories, in the paper on A∞–categories by Lyubashenko. It seems that it is a good idea to generalize his definition first to the case of an iterated monoidal base category and then to define V–(n + 1)–categories as categories enriched over V–n–Cat, the (k−n)–fold monoidal strict (n+1)–category of V–n–categories where k<n ∈ N. We show that for V k–fold monoidal the structure of a (k−n)–fold monoidal strict (n + 1)–category is possessed by V–n–Cat. / Ph. D.

We consider the regularization of some equivariant Euler classes of certain infinite-dimensional vector bundles over a finite-dimensional manifold M using the framework of zeta-regularized products [35, 53, 59]. An example of such a regularization is the Atiyah–Witten regularization of the T-equivariant Euler class of the normal bundle v(TM) of M in the free loop space LM [2]. In this thesis, we propose a new regularization procedure — W-regularization — which can be shown to reduce to the Atiyah–Witten regularization when applied to the case of v(TM). This new regularization yields a new multiplicative genus (in the sense of Hirzebruch [26]) — the ^Γ-genus — when applied to the more general case of a complex spin vector bundle of complex rank ≥ 2 over M, as opposed to the case of the complexification of TM for the Atiyah–Witten regularization. Some of its properties are investigated and some tantalizing connections to other areas of mathematics are also discussed. We also consider the application of W-regularization to the regularization of T²- equivariant Euler classes associated to the case of the double free loop space LLM. We find that the theory of zeta-regularized products, as set out by Jorgenson–Lang [35], Quine et al [53] and Voros [59], amongst others, provides a good framework for comparing the regularizations that have been considered so far. In particular, it reveals relations between some of the genera that appeared in elliptic cohomology, allowing us to clarify and prove an assertion of Liu [44] on the ˆΘ-genus, as well as to recover the Witten genus. The ^Γ₂-genus, a new genus generated by a function based on Barnes’ double gamma function [5, 6], is also derived in a similar way to the ^Γ-genus. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

We consider the regularization of some equivariant Euler classes of certain infinite-dimensional vector bundles over a finite-dimensional manifold M using the framework of zeta-regularized products [35, 53, 59]. An example of such a regularization is the Atiyah–Witten regularization of the T-equivariant Euler class of the normal bundle v(TM) of M in the free loop space LM [2]. In this thesis, we propose a new regularization procedure — W-regularization — which can be shown to reduce to the Atiyah–Witten regularization when applied to the case of v(TM). This new regularization yields a new multiplicative genus (in the sense of Hirzebruch [26]) — the ^Γ-genus — when applied to the more general case of a complex spin vector bundle of complex rank ≥ 2 over M, as opposed to the case of the complexification of TM for the Atiyah–Witten regularization. Some of its properties are investigated and some tantalizing connections to other areas of mathematics are also discussed. We also consider the application of W-regularization to the regularization of T²- equivariant Euler classes associated to the case of the double free loop space LLM. We find that the theory of zeta-regularized products, as set out by Jorgenson–Lang [35], Quine et al [53] and Voros [59], amongst others, provides a good framework for comparing the regularizations that have been considered so far. In particular, it reveals relations between some of the genera that appeared in elliptic cohomology, allowing us to clarify and prove an assertion of Liu [44] on the ˆΘ-genus, as well as to recover the Witten genus. The ^Γ₂-genus, a new genus generated by a function based on Barnes’ double gamma function [5, 6], is also derived in a similar way to the ^Γ-genus. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

We give a definition of the scanning map for configuration spaces that is equivariant under the action of the diffeomorphism group of the underlying manifold. We use this to extend the Bödigheimer-Madsen result for the stable splittings of the Borel constructions of certain mapping spaces from compact Lie group actions to all smooth actions. Moreover, we construct a stable splitting of configuration spaces which is equivariant under smooth group actions, completing a zig-zag of equivariant stable homotopy equivalences between mapping spaces and certain wedge sums of spaces. Finally we generalise these results to configuration spaces with twisted labels (labels in a fibre bundle subject to certain conditions) and extend the Bödigheimer-Madsen result to more mapping spaces.

This thesis consists of four papers on the topics of free loop spaces, Koszul duality and A∞-algebras.  In Paper I we consider a definition of differential operators for noncommutative algebras. This definition is inspired by the connections between differential operators of commutative algebras, L∞-algebras and BV-algebras. We show that the definition is reasonable by establishing results that are analoguous to results in the commutative case. As a by-product of this definition we also obtain definitions for noncommutative versions of Gerstenhaber and BV-algebras.  In Paper II we calculate the free loop space homology of (n-1)-connected manifolds of dimension of at least 3n-2. The Chas-Sullivan loop product and the loop bracket are calculated. Over a field of characteristic zero the BV-operator is determined as well. Explicit expressions for the Betti numbers are also established, showing that they grow exponentially.  In Paper III we restrict our coefficients to a field of characteristic 2. We study the Dyer-Lashof operations that exist on free loop space homology in this case. Explicit calculations are carried out for manifolds that are connected sums of products of spheres.  In Paper IV we extend the Koszul duality methods used in Paper II by incorporating A∞-algebras and A∞-coalgebras. This extension of Koszul duality enables us to compute free loop space homology of manifolds that are not necessarily formal and coformal. As an example we carry out the computations for a non-formal simply connected 7-manifold. / Denna avhandling består av fyra artiklar inom ämnena fria öglerum, Koszuldualitet och A∞-algebror. I Artikel I behandlar vi en definition av differentialoperatorer för ickekommutativa algebror. Denna definition är inspirerad av kopplingar mellan differentialoperatorer för kommutativa algebror, L∞-algebror och BV-algebror. Vi visar att definitionen är rimlig genom att etablera resultat som är analoga med resultat i det kommutativa fallet. Som en biprodukt får vi också definitioner för ickekommutativa varianter av Gerstenhaber och BV-algebror. I Artikel II beräknar vi den fria öglerumshomologin av (n-1)-sammanhängande mångfalder av dimension minst 3n-2. Chas-Sullivans ögleprodukt och öglehake beräknas. Över en kropp av karakteristik noll beräknas även BV-operatorn. Explicita uttryck för Bettitalen fastställs också, vilka visar att de växer exponentiellt. I Artikel III begränsar vi koefficienterna till en kropp av karakteristik 2. Vi studerar Dyer- Lashofoperationer som existerar på den fria öglerumshomologin i detta fall. Explicita beräkningar görs för mångfalder som är sammanhängande summor av produkter av sfärer. I Artikel IV utvidgar vi Koszuldualitetmetoden som används i Artikel II genom att inkorporera A∞-algebror och A∞-koalgebror. Denna utvidgning av Koszuldualitet gör det möjligt att beräkna fri öglerumshomologi för mångfalder som inte nödvändigtvis är formella och koformella. Som ett exempel utför vi beräkningar för en ickeformell enkelt sammanhängande 7-mångfald. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.</p>

The Chow/Van der Waerden approach to algebraic cycles via resultants is elaborated and used to give a purely algebraic proof for the algebraicity of the complex suspension over arbitrary fields. The algebraicity of the join pairing on Chow varieties then follows over the complex numbers. The approach implies a more algebraic proof of Lawson´s complex suspension theorem in characteristic 0. The continuity of the action of the linear isometries operad on the group completion of the stable Chow variety is a consequence. Further Hoyt´s proof of the independence of the algebraic-continuous homeomorphism type of Chow varieties on embeddings is rectified and worked out over arbitrary fields.

O princípio de reconhecimento de espaços de $\\infty$-laços é que o funtor $\\Omega^\\infty:\\textttightarrow \\mathcal E^\\infty[\\texttt]$ dado por $\\Omega^\\infty Y_\\bullet=\\text_{\\bullet\\shortrightarrow\\infty}\\Omega^\\bullet Y_\\bullet$ induz uma equivalência entre a categoria homotópica de espectros conectivos e a categoria homotópica de $\\mathcal E^\\infty$-álgebras grouplike para qualquer resolução cofibrante $\\mathcal E^\\infty$ do operad $\\mathcal Com$ de monóides comutativos. Nesta tese é provado um princípio de reconhecimento de 2-espaços de $N$-laços para $2<N\\leq\\infty$. Quando $N=\\infty$ esse princípio afirma o seguinte: Um espectro relativo é um par de espectros $B_\\bullet$ e $Y_\\bullet$ equipados com uma sequências de aplicações pontuadas $\\iota_\\bullet:B_\\bulletightarrow Y_{\\bullet+1}$ compatíveis com as estruturas de espectros. Um espectro relativo é conectivo se o par de espectros subjacentes forem conectivos. Denotamos a categoria de espectros relativos por $\\texttt^ earrow$ e de espectros relativos conectivos por $\\texttt^ earrow_0$. Um $2E_\\infty$-operad é uma resolução cofibrante $\\mathcal E_2^\\infty$ do 2-operad $\\mathcal Com^\\shortrightarrow$ de homomorfismos de monóides comutativos. Uma $\\mathcal E^\\infty_2$-álgebra $(X_c,X_o)$ é grouplike se $X_c$ e $X_o$ forem grouplike. Denotamos a categoria de $\\mathcal E^\\infty_2$-álgebras por $\\mathcal E^\\infty_2[\\texttt]$ e a categoria de $\\mathcal E^\\infty_2$-álgebras grouplike por $\\mathcal E^\\infty_2[\\texttt]_$. O 2-espaço de $\\infty$-laços de um espectro relativo é o par de espaços $\\Omega^\\infty_2\\iota_\\bullet:=\\text_{\\bullet\\shortrightarrow\\infty}(\\Omega^\\bullet Y_\\bullet,\\Omega^{\\bullet}_{\\text} \\iota_\\bullet)$. Temos que as imagens do funtor $\\Omega^\\infty_2$ admitem uma estrutura natural de $\\mathcal E^\\infty_2$-álgebra, logo $\\Omega^\\infty_2$ define um funtor $\\texttt^ earrowightarrow \\mathcal E^\\infty_2[\\texttt]$. Existe um funtor $B^\\infty_2:\\mathcal E^\\infty_2[\\texttt]ightarrow \\texttt^ earrow$ e uma adjunção $(\\mathbb L B^\\infty_2\\dashv\\mathbb R\\Omega^\\infty_2)$ entre as categorias homotópicas $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]$ e $\\mathcal Ho\\texttt^ earrow$ que induzem uma equivalência entre as categorias homotópicas $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]_$ e $\\mathcal Ho\\texttt^ earrow_0$. / The recognition principle of $\\infty$-loop spaces is that the functor $\\Omega^\\infty:\\textttightarrow \\mathcal E^\\infty[\\texttt]$ defined by $\\Omega^\\infty Y_\\bullet=\\text_{\\bullet\\shortrightarrow\\infty}\\Omega^\\bullet Y_\\bullet$ induces an equivalence between the homotopy category of connective spectra and the homotopy category of grouplike $\\mathcal E^\\infty$-algebras for any cofibrant resolution $\\mathcal E^\\infty$ of the commutative monoid operad $\\mathcal Com$. In this thesis a relative recognition principle of $N$-loop 2-spaces is proved for $2<N\\leq\\infty$. For $N=\\infty$ this principle states the following: A relative spectrum is a pair of spectra $B_\\bullet$ and $Y_\\bullet$ equipped with a sequence of pointed maps $\\iota_\\bullet:B_\\bulletightarrow Y_{\\bullet+1}$ compatible with the spectrum structures. A relative spectrum is connective if the underlying pair of spectra are connective. The category of relative spectra is denoted by $\\texttt^ earrow$ and the category of connective relative spectra by $\\texttt^ earrow_0$. A $2E_\\infty$-operad is a cofibrant resolution $\\mathcal E_2^\\infty$ of the commutative monoid homomorphism 2-operad $\\mathcal Com^\\shortrightarrow$. An $\\mathcal E^\\infty_2$-algebra $(X_c,X_o)$ is grouplike if $X_c$ and $X_o$ are grouplike. The category of $\\mathcal E^\\infty_2$-algebras is denoted by $\\mathcal E^\\infty_2[\\texttt]$ and the category of grouplike $\\mathcal E^\\infty_2$-algebras by $\\mathcal E^\\infty_2[\\texttt]_$. The $\\infty$-loop 2-space of a relative spectrum is the pair of pointed spaces $\\Omega^\\infty_2\\iota_\\bullet:=\\text_{\\bullet\\shortrightarrow\\infty}(\\Omega^\\bullet Y_\\bullet,\\Omega_{\\text}^{\\bullet} \\iota_\\bullet)$. The images of the functor $\\Omega^\\infty_2$ admit an $\\mathcal E^\\infty_2$-algebra structure, therefore $\\Omega^\\infty_2$ defines a functor $\\texttt^ earrowightarrow \\mathcal E^\\infty_2[\\texttt]$. The infinite relative recognition principle is that there is a functor $B^\\infty_2:\\mathcal E^\\infty_2[\\texttt]ightarrow \\texttt^ earrow$ and a derived adjunction $(\\mathbb L B^\\infty_2\\dashv\\mathbb R\\Omega^\\infty_2)$ between the homotopy categories $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]$ and $\\mathcal Ho\\texttt^ earrow$ that induce an equivalence beteween the homotopy categories $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]_$ and $\\mathcal Ho\\texttt^ earrow_0$.

O princípio de reconhecimento de espaços de $\\infty$-laços é que o funtor $\\Omega^\\infty:\\textttightarrow \\mathcal E^\\infty[\\texttt]$ dado por $\\Omega^\\infty Y_\\bullet=\\text_{\\bullet\\shortrightarrow\\infty}\\Omega^\\bullet Y_\\bullet$ induz uma equivalência entre a categoria homotópica de espectros conectivos e a categoria homotópica de $\\mathcal E^\\infty$-álgebras grouplike para qualquer resolução cofibrante $\\mathcal E^\\infty$ do operad $\\mathcal Com$ de monóides comutativos. Nesta tese é provado um princípio de reconhecimento de 2-espaços de $N$-laços para $2<N\\leq\\infty$. Quando $N=\\infty$ esse princípio afirma o seguinte: Um espectro relativo é um par de espectros $B_\\bullet$ e $Y_\\bullet$ equipados com uma sequências de aplicações pontuadas $\\iota_\\bullet:B_\\bulletightarrow Y_{\\bullet+1}$ compatíveis com as estruturas de espectros. Um espectro relativo é conectivo se o par de espectros subjacentes forem conectivos. Denotamos a categoria de espectros relativos por $\\texttt^ earrow$ e de espectros relativos conectivos por $\\texttt^ earrow_0$. Um $2E_\\infty$-operad é uma resolução cofibrante $\\mathcal E_2^\\infty$ do 2-operad $\\mathcal Com^\\shortrightarrow$ de homomorfismos de monóides comutativos. Uma $\\mathcal E^\\infty_2$-álgebra $(X_c,X_o)$ é grouplike se $X_c$ e $X_o$ forem grouplike. Denotamos a categoria de $\\mathcal E^\\infty_2$-álgebras por $\\mathcal E^\\infty_2[\\texttt]$ e a categoria de $\\mathcal E^\\infty_2$-álgebras grouplike por $\\mathcal E^\\infty_2[\\texttt]_$. O 2-espaço de $\\infty$-laços de um espectro relativo é o par de espaços $\\Omega^\\infty_2\\iota_\\bullet:=\\text_{\\bullet\\shortrightarrow\\infty}(\\Omega^\\bullet Y_\\bullet,\\Omega^{\\bullet}_{\\text} \\iota_\\bullet)$. Temos que as imagens do funtor $\\Omega^\\infty_2$ admitem uma estrutura natural de $\\mathcal E^\\infty_2$-álgebra, logo $\\Omega^\\infty_2$ define um funtor $\\texttt^ earrowightarrow \\mathcal E^\\infty_2[\\texttt]$. Existe um funtor $B^\\infty_2:\\mathcal E^\\infty_2[\\texttt]ightarrow \\texttt^ earrow$ e uma adjunção $(\\mathbb L B^\\infty_2\\dashv\\mathbb R\\Omega^\\infty_2)$ entre as categorias homotópicas $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]$ e $\\mathcal Ho\\texttt^ earrow$ que induzem uma equivalência entre as categorias homotópicas $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]_$ e $\\mathcal Ho\\texttt^ earrow_0$. / The recognition principle of $\\infty$-loop spaces is that the functor $\\Omega^\\infty:\\textttightarrow \\mathcal E^\\infty[\\texttt]$ defined by $\\Omega^\\infty Y_\\bullet=\\text_{\\bullet\\shortrightarrow\\infty}\\Omega^\\bullet Y_\\bullet$ induces an equivalence between the homotopy category of connective spectra and the homotopy category of grouplike $\\mathcal E^\\infty$-algebras for any cofibrant resolution $\\mathcal E^\\infty$ of the commutative monoid operad $\\mathcal Com$. In this thesis a relative recognition principle of $N$-loop 2-spaces is proved for $2<N\\leq\\infty$. For $N=\\infty$ this principle states the following: A relative spectrum is a pair of spectra $B_\\bullet$ and $Y_\\bullet$ equipped with a sequence of pointed maps $\\iota_\\bullet:B_\\bulletightarrow Y_{\\bullet+1}$ compatible with the spectrum structures. A relative spectrum is connective if the underlying pair of spectra are connective. The category of relative spectra is denoted by $\\texttt^ earrow$ and the category of connective relative spectra by $\\texttt^ earrow_0$. A $2E_\\infty$-operad is a cofibrant resolution $\\mathcal E_2^\\infty$ of the commutative monoid homomorphism 2-operad $\\mathcal Com^\\shortrightarrow$. An $\\mathcal E^\\infty_2$-algebra $(X_c,X_o)$ is grouplike if $X_c$ and $X_o$ are grouplike. The category of $\\mathcal E^\\infty_2$-algebras is denoted by $\\mathcal E^\\infty_2[\\texttt]$ and the category of grouplike $\\mathcal E^\\infty_2$-algebras by $\\mathcal E^\\infty_2[\\texttt]_$. The $\\infty$-loop 2-space of a relative spectrum is the pair of pointed spaces $\\Omega^\\infty_2\\iota_\\bullet:=\\text_{\\bullet\\shortrightarrow\\infty}(\\Omega^\\bullet Y_\\bullet,\\Omega_{\\text}^{\\bullet} \\iota_\\bullet)$. The images of the functor $\\Omega^\\infty_2$ admit an $\\mathcal E^\\infty_2$-algebra structure, therefore $\\Omega^\\infty_2$ defines a functor $\\texttt^ earrowightarrow \\mathcal E^\\infty_2[\\texttt]$. The infinite relative recognition principle is that there is a functor $B^\\infty_2:\\mathcal E^\\infty_2[\\texttt]ightarrow \\texttt^ earrow$ and a derived adjunction $(\\mathbb L B^\\infty_2\\dashv\\mathbb R\\Omega^\\infty_2)$ between the homotopy categories $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]$ and $\\mathcal Ho\\texttt^ earrow$ that induce an equivalence beteween the homotopy categories $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]_$ and $\\mathcal Ho\\texttt^ earrow_0$.