"On the Sphere and Cylinder" is a mathematical work by the ancient Greek philosopher and mathematician Archimedes. Written in the 3rd century BC, the treatise explores the geometric properties of spheres and cylinders, deriving formulas related to their volumes and surface areas. In the text, Archimedes examines the relationships between these shapes, showcasing his groundbreaking methods in geometry.

A **simple polytope** is a type of polytope characterized by certain geometric properties. Specifically, it is defined as a convex polytope in which every face is a simplex. In more technical terms, a polytope is called simple if at each vertex, exactly \(d\) edges (where \(d\) is the dimension of the polytope) meet.

The Theorem of the Gnomon is a mathematical concept related to geometric figures, particularly in the context of areas. Although it is not as commonly referenced as other theorems, it essentially deals with the relationship between certain geometric shapes, particularly in relation to squares and rectangles. The term "gnomon" refers to a shape that, when added to a particular figure, results in a new figure that is similar to the original.

In the context of geometry, particularly when discussing triangles, "straight lines" generally refer to the sides of a triangle. A triangle is defined by three straight lines that connect three points, known as vertices, in a two-dimensional plane. These straight lines meet the following criteria: 1. **Straightness**: Each side is a straight line segment connecting two vertices. 2. **Consecutive**: Each side is adjacent to two other sides, forming the perimeter of the triangle.

Cellular homology is a tool in algebraic topology that allows for the computation of homology groups of a topological space by using a cellular structure derived from a CW-complex. A CW-complex is a kind of topological space that is built up from basic building blocks called cells, which are homeomorphic to open disks in Euclidean space, glued together in a specific way.

The Hodge conjecture is a fundamental statement in algebraic geometry and topology that relates the topology of a non-singular projective algebraic manifold to its algebraic cycles. Formulated by W.V. Hodge in the mid-20th century, the conjecture suggests that certain classes of cohomology groups of a projective algebraic variety have a specific geometric interpretation.

Poincaré duality is a fundamental theorem in algebraic topology that describes a duality relationship between certain topological spaces, particularly manifolds, and their cohomology groups. Named after the French mathematician Henri Poincaré, the theorem specifically applies to compact, oriented manifolds.

In homotopy theory, a branch of topology, theorems often deal with properties of spaces and maps (functions between spaces) that remain invariant under continuous deformations, such as stretching and bending, but not tearing or gluing.

In topology, a cofibration is a specific type of map between topological spaces that satisfies certain conditions. Cofibrations play a crucial role in homotopy theory and the study of fibration and cofibration sequences. They are often defined in terms of the homotopy extension property. ### Definition: A map \( i : A \to X \) is called a **cofibration** if it satisfies the homotopy extension property with respect to any space \( Y \).

The Cotangent complex is a fundamental construction in algebraic geometry and homotopy theory, especially within the context of derived algebraic geometry. It can be seen as a tool to study the deformation theory of schemes and their morphisms.

Simple homotopy theory is a branch of algebraic topology that provides a way to study the properties of topological spaces through the lens of homotopy equivalence. It is particularly concerned with the study of CW complexes and involves a concept known as simple homotopy equivalence. ### Key Concepts 1. **Homotopy**: In general, homotopy is a relation between continuous functions, where two functions are considered equivalent if one can be transformed into the other through continuous deformation.

The Generalized Poincaré Conjecture extends the classical Poincaré Conjecture, which is a statement about the topology of 3-dimensional manifolds. The original Poincaré Conjecture, proposed by Henri Poincaré in 1904, asserts that any simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Homotopical connectivity is a concept from algebraic topology, a branch of mathematics that studies topological spaces through the lens of homotopy theory. It provides a way to classify topological spaces based on their "connectedness" in a homotopical sense. In more detail, homotopical connectivity can be understood through the following concepts: 1. **Connectedness**: A topological space is called connected if it cannot be divided into two disjoint open sets.

In topology, the localization of a topological space is a method of constructing a new topological space from an existing one by focusing on a particular subset of the original space. The concept of localization can be understood in several contexts, such as the localization of rings or sheaves, but here I will outline the localization of a topological space itself, particularly in algebraic topology. ### 1.

In category theory, an \( N \)-group is a concept that extends the notion of groups to a more general framework, particularly in the context of higher-dimensional algebra. The term "N-group" can refer to different concepts depending on the specific area of study, but it is commonly associated with the study of higher categories and homotopy theory.

The Novikov conjecture is a significant hypothesis in the field of topology and geometry, particularly concerning the relationships between the algebraic topology of manifolds and their geometric structure. It was proposed by the Russian mathematician Sergei Novikov in the 1970s. At its core, the Novikov conjecture deals with the higher dimensional homotopy theory, specifically the relationship between the homotopy type of a manifold and the groups of self-homotopy equivalences of the manifold.

In mathematics, particularly in the field of algebraic topology, a **simplicial space** is a topological space that is equipped with a simplicial structure. More specifically, a simplicial space is a contravariant functor from the simplex category, which comprises simplices of various dimensions and their face and degeneracy maps, to the category of topological spaces.

The Whitehead product is a concept from algebraic topology, specifically in the context of algebraic K-theory and homotopy theory. It is named after the mathematician G. W. Whitehead and plays a significant role in the study of higher homotopy groups and the structure of loop spaces. In general, the Whitehead product is a binary operation that can be defined on the homotopy groups of a space.

The Atiyah–Hirzebruch spectral sequence is an important tool in algebraic topology, specifically in the computation of homotopy groups and cohomology theories. It provides a way to calculate the homology or cohomology of a space using a spectral sequence that is associated with a specific filtration. The original context for the spectral sequence primarily relates to complex vector bundles and characteristic classes.

A **circle bundle** is a specific type of fiber bundle in differential geometry, where the fibers are circles \( S^1 \).