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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 …