# Tagged Questions

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

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### Orientation on a manifold as a sheaf

I am thinking about orientation of a connected manifold $M$ of dim $n$ as a sheaf. There are two definitions I could use, the first is the sheaf associated to the presheaf $$U\mapsto H_n(M,M-U;R).$$ ...
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### Computational Topology and Lie Group Theory [closed]

I study Machine Learning and my limited background in math is enough to understand all the popular algorithms and methods. However, recently, Topology has been successfully applied to Data Analysis ...
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### “the only odd dimensional spheres with a unique smooth structure are $S^1$, $S^3$, $S^5$, $S^{61}$”

This (long) paper, Guozhen Wang, Zhouli Xu. "On the uniqueness of the smooth structure of the 61-sphere." arXiv:1601.02184 [math.AT]. proves that the only odd dimensional spheres with a ...
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### Is the smash product of two Moore spaces again a Moore space?

Write $M(G,n)$ for the Moore space with $\tilde{H}_\ast(M(G,n);\mathbb{Z})$ naturally isomorphic to $G$ concentrated in degree $n$. Now fix finitely generated (Abelian) groups $G$ and $H$ and ...
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### Whitney and Tensor product structures on $BU$.

I have a question regarding the 2 H-space structures on $BU$. My current understanding (which may not be correct!) is detailed below. $BU$ admits two H-space structures described as follows: Let ...
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### Contractibility is a Weaker Notion than Deformation Retract to a Point [duplicate]

This is problem 6 in Chapter 0 of Hatcher's Algebraic Topology. Let $X$ be the subspace of $\mathbf R^2$ consisting of the horizontal line segment $[0, 1]\times \{0\}$ together with the vertical ...
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### Proof of $BGl(n)\simeq Gr(n,\infty)$

What is a direct proof for the existence of a weak homotopy equivalence between the Grassmanian $Gr(n):=Gr(n,\infty)$ and the classifying space $BGL(n)$ of $GL(n)$? They both represent the ...
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### compute the homology groups

Consider the complex $M$, which is the union of three triangles $v_1v_2v_5, v_1v_5v_4, v_4v_5v_3$ and the line segment $v_2v_3$. Compute the homology groups $H_1(M)$ and $H_2(M)$. Is ...
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### Van Kampen theorem application in a simple three-holed figure

The purpose of this question is to understand the computations to get the expression of the fundamental group in a simple case using the Van Kampen theorem. Let $X$ be the three holes object ...
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### Parameterizing the boundary of a closed set in R³

I've got some closed set in R³ defined in the form of a vector equation that's a function of three parameters: (x,y,z) = (f[u,v,w], g[u,v,w], h[u,v,w]) The ...
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### What is the space obtained by identifying boundary $\mathbb T^2$ of a solid torus

By identifying boundary of solid $\mathbb T^2$, one obtains a 3-manifold, but what is the space "looks like"? For example, can we understand it through Heegaard splitting? More generally I want to ask ...
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### Atiyah-Segal axioms for TQFT [closed]

Could someone explain the importance of the Atiyah-Segal axioms for TQFT? Why is this studied by mathematicians, why is it interesting or useful?
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### Topological surface covered by hexagons and heptagons

I've found an interesting exercice that I don't know how to approach. It goes like this. We have a topological space which is Hausdorff, compact, connected and locally homeomorphic to ...
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### Defining Induced map of homology groups of torus

In this question [1]: Induced map of homology groups of torus I couldn't understand the induced map on $H_1(X)$, $f_∗:H_1(X)→H_1(X)$ look like, how we define this map ?
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### Permuted action of the ramified covering

Let $f:E\rightarrow S^2$ be a ramified covering of degree n, and let $t_1,t_2,..t_m$ be all its points of ramifications. Pick a point $t\in S^2$ distinct from all $t_i$ and connect it with the points ...
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### Introduction to homology of simplitial complex.

I know the outline of homology theory of simplitial complex and be able to trianglulate some simple figures and compute the homology groups of them, but don't know theoretial details such as what kind ...
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### Glueing manifolds with boundaries and Seifert-Van Kampen theorem

I've seen many times the following application of the SVK theorem: Let $M$ and $N$ two smooth $n$-manifolds ($n\ge 3$) with boundary and suppose that they have the same boundary $B$. Now, after ...
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### Hatcher Corollary 4.12

Just to ask a quick question regarding a corollary 4.12 in Hatcher: "A CW pair $(X,A)$ is n-connected if all the cells in $X-A$ have dimension greater than $n$. In particularly the pair $(X,X^n)$ is ...
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### Zeroth homotopy group: what exactly is it?

What are the elements in the zeroth homotopy group? Also, why does $\pi_0(X)=0$ imply that the space is path-connected? Thanks for the help. I find that zeroth homotopy groups are rarely discussed in ...
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### Getting the most general form of Mayer-Vietoris from the axioms of homology

I'd like to derive the most general form of the Mayer-Vietoris sequence from the Eilenberg-Steenrod axioms for homology (in particular: I do not want to use the definition of $H_\ast(X)$ in terms of ...
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### The fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed

Determine whether the fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed is trivial, infinite cyclic, or isomorphic to the figure eight space. I found this answer: ...
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### Equivariant cohomology via equivariant sheaves

Ordinary cohomology of topological space $X$ are known to be the cohomology of constant sheaf. Question Is there analogous description for equivariant cohomology? More precisely. Consider category ...
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### Showing $\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}/(1,-1,0)\mathbb{Z}+(0,1,1)\mathbb{Z}+(1,0,-1)\mathbb{Z}\cong \mathbb{Z}$

I have to prove that if $V_K = \{v_0, v_1, v_2\}$ and $K = \{\{v_0\}, \{v_1\}, \{v_2\}, \{v_0, v_1\}, \{v_0, v_2\}, \{v_1, v_2\}\}$ then $H_q(K, \mathbb{Z})\cong \mathbb{Z}$ for $q = 0, 1$. Already ...
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### Why did $Ext$ appear to make the sequence exact after taking its dual?

The above question is about the exact sequence in the bottom of the following figure from p196 of Hatcher's text. After taking the dual of the original short exact sequence, $Ext$ comes in at the end ...
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### Counterexamples on homotopy equivalence and infinite product [closed]

Let $(X,a),(Y,b)$ be pointed spaces and $f:(X,a)\rightarrow (Y,b)$ be a continuous function. If the natural homomorphism $f_*:\pi_1(X,a)\rightarrow \pi_1(Y,b)$ is a group isomorphism, is $f$ ...
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### Interpretation for the curvature and monodromy of a connection - Reality check

Let $P \to M$ be a principal $G$-bundle with connection form $\omega \in \Omega^1(P,\mathfrak{g})$. Here are the statements I'm basing my viewpoint on: A connection is flat (vanishing ...
In exercise 1.3.28 in Hatcher's Algebraic Topology, we are asked to show that for a covering space action of a group $G$ on a simply-connected space $Y$, $\pi_1(Y/G)$ is isomorphic to $G$. This is a ...
### Is $X$ a subset of $CX$?
In Spanier's, Algebraic Topology, he writes: "A topological pair $(X,A)$ consists of a topological space $X$ and a subspace $A \subset X$." In a question at the end of the section he asks a ...