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Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

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Circle functions and its fibres

How can i formally deduce from the definition of homotopy that if a circle function $f: S^1 \rightarrow S^1$ is homotopic to the n-th power function $z \mapsto z^n$, then its mapping wraps around $S^1$... user avatar ultra

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1vote0answers17views

Showing that the identity component of a linear algebraic group is a closed normal subgroup

Recall that a topological space is called connected if it cannot be expressed as the unsion of two disjoint non-empty open subsets. Every topological space admits a partition into its connected ... user avatar user12345

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0votes0answers34views

When is a space $X$ a retract of $\mathbb{R}^2$?

Problem: Let $$X_K = ([0, 1] \times \{0, 1\}) \cup (\{0\} \times [0, 1]) \cup (\bigcup_{x \in K} \{x\} \times [0, x]),$$ where $K \subseteq \mathbb{R}$ is a countable set. Give the necessary and ... user avatar the_dude

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Triangulation of Mobius Strip to find the fundamental group

I am trying to show from image below that the fundamental group of the Mobius strip is $\pi_1(\text{Mobius strip})\cong\mathbb{Z}$ by finding the maximal tree of the Mobius strip. I know how to do ... user avatar Cassem02

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Area of Complete Hyperbolic surface

I'm studying the area of hyperbolic surfaces and have reached a proposition that is not understandable.I really appreciate it if you could help me with it. Proposition: A complete hyperbolic surface F ... user avatar mariotic_metric

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2votes0answers31views

When is the topological space an absolute extensor for normal spaces?

Problem: Let $$X_K = ([0, 1] \times \{0, 1\}) \cup (\{0\} \times [0, 1]) \cup (\bigcup_{x \in K} \{x\} \times [0, x]),$$ where $K \subseteq \mathbb{R}$ is a countable set. Give the necessary and ... user avatar the_dude

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Algebraic Topology: Question on meaning of $f_t \vert A$ where $A \subseteq X$?

I am goingAlgebraic Topology: Question on meaning of $f_t \vert A$ where $A \subseteq X$? through Allen Hatcher's Algebraic Topology book and noticed the use of $f_t \vert A$ for some subspace $A$ of ... user avatar allyouneed

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Is de Rham cohomology a generalized cohomology?

I guess that the de Rham cohomology $H^{*}_{dR}(M)$ of a smooth manifold $M$ is naturally isomorphic to a cellular cohomology with coefficient $\mathbb{R}$. To do this, we need only check that de Rham ... user avatar s.h

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Reeb Graph of Topological Space and Lemma about Induced map Between Homologies

I am a physicist following a course in Topological Data Analysis (MasterMath), and I need to prove the following lemma about the graphs. I know I am supposed to write what I have tried, but honestly I ... user avatar George Smyridis

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Torus and projective space

Let $X = T^2 \sharp P^2 = 3P^2$. If a remove an interior point of $X$, I can retract the space onto the boundary. This new space is the wedge of three $S^1$? user avatar hal97

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Euler characteristic of the join of a simplex and the boundary of another simplex

Let $\Delta^n$ be the abstract simplicial complex on $n + 1$ vertices where every nonempty subset of vertices is a face. Let $\partial \Delta^n$ be the abstract simplicial complex on $n + 1$ vertices ... user avatar Colin Tan

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5votes2answers89views

Is the space of distinct triples homeomorphic to a union of products?

$\newcommand{\S}{\mathbb{S}^1}$Let $M=\{(x,y,z) \in (\S)^3 \, |\,\, x,y,z \,\,\text{are distinct}\}$. Is $M$ homeomorphic to a finite union of products of one-dimensional manifolds? I think $M$ is ... user avatar Asaf Shachar

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Topology of Kaehler manifold and Curvature

I wonder what kind area studies the relation between the differential geometric (local invariant) date and the global(topological) nature of the Kaehler manifold? I like the way, 1. Bochner's ... user avatar Mary Susy

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Showing that a 4-manifold obtained by attaching a 2-handle is simply-connected

I am reading Lemma 2.1 of this paper () and I can't see why $W$ is simply-connected. Here is the situation: Let $K$ be a ribbon knot in $S^3$; it bounds a ribbon ... user avatar user302934

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Find an error in computation of $H^n(RP^\infty;\mathbb{Z}_2)$ using cellular structure of the real projective space

I'm following Milnor and Stasheff, Characteristic classes, now it leaves without proof the computation of $H^*(RP^\infty;Z_2)$. I know about the computation in Hatcher's book, nevertheless I thought I ... user avatar AOJIDSOeoi

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