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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 $S^1$... general-topology algebraic-topology- 13
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 ... linear-algebra algebraic-topology- 125
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 ... general-topology algebraic-topology- 145
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 ... algebraic-topology homology-cohomology homotopy-theory- 35
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 ... differential-geometry algebraic-topology hyperbolic-geometry- 1
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 ... general-topology algebraic-topology- 145
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 ... general-topology algebraic-topology- 1
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 ... algebraic-topology differential-topology smooth-manifolds differential-forms de-rham-cohomology- 91
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 ... algebraic-topology topological-data-analysis- 429
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 $S^1$? algebraic-topology- 353
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 ... algebraic-topology simplicial-complex- 169
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 ... general-topology algebraic-topology differential-topology topological-groups symmetry- 23.4k
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 ... differential-geometry algebraic-geometry partial-differential-equations algebraic-topology- 11
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 ... algebraic-topology manifolds fundamental-groups surgery-theory 4-manifolds- 2,130
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 ... algebraic-topology homology-cohomology fake-proofs- 23
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