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Algebra prerequisites for Homology Theory

I am a first year graduate student in Mathematics. I am planning to take a graduate course on Homology Theory. My background is Point Set Topology (material covered in Part 1 of Munkres) and the ...

Is a simply connected set connected?

A set $X$ is considered connected if there is no separation of the set $X$ into disjoint sets $A,B$ such that $X = A \cup B$, where neither sets ($A$,$B$) contain limit points of each other. Now a ...

When is a regular map a covering map

Let M, N be two manifolds of the same dimension A map from M to N is regular provided its tangent map is one to one. A map from M to N is a covering map provided each point in N has a neighborhood …

Trying to Understand Lefschetz Pencils

I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) better, tho I would appreciate insights on condition i), and in general. A Lefschetz pencil on a $4-$ manifold $X$ is a …

Anabelian geometry study materials?

I want to study anabelian geometry, but unfortunately I'm having difficulties in finding some materials about it. If you could offer me some books/papers/articles I would be glad.

Technology for various models of spectra

There are a couple different models for spectra, or constructions of the categories of spectra that have the desired properties (homotopically and otherwise). The construction of the Categories of ...

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle …

Infinite loop spaces

Let $X, Y$ be infinite loop spaces: $X = QA$ and $Y = QB$, where $A,B$ are connected topological spaces, and $Q$ stands for $\Omega^\infty S^\infty.$ Let $f:X \to Y$ be a continuous map such that ...

Pontryagin numbers on manifolds with an $S^1$-action

Let $M$ is a smooth compact manifold with an $S^1$-action with isolated fixed points. Suppose the representation of $S^1$ at tangent spaces at all fixed points is known. Can one then find all ...

Cohomology groups for the following pair $(X,A)$

Let $X=S^1\times D^2$, and let $A=\{(z^k,z)\mid z\in S^1\}\subset X$. Calculate the groups and homomorphisms in the cohomology of the exact sequence of the pair $(X,A)$. I know that theorically one …

Specifically, the question says to consider the torus $T$ as a square with the usual identifications, with two opposite boundary edges labelled $a$ and the other two edges labelled $b$, and consider …