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

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5
votes
1answer
50 views

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 $$U\mapsto H_n(M,M-U;R).$$ ...
3
votes
1answer
70 views

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 ...
26
votes
1answer
346 views

“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 ...
3
votes
1answer
40 views

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 ...
1
vote
0answers
37 views

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 ...
0
votes
0answers
30 views

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 ...
0
votes
1answer
56 views

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 ...
0
votes
1answer
46 views

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 ...
2
votes
1answer
42 views

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 ...
0
votes
0answers
12 views

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 ...
3
votes
1answer
53 views

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 ...
3
votes
0answers
71 views

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?
1
vote
1answer
57 views

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 ...
0
votes
0answers
45 views

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 ?
0
votes
0answers
14 views

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 ...
1
vote
0answers
33 views

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 ...
4
votes
1answer
53 views

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 ...
1
vote
2answers
56 views

Fundamental group of a manifold minus a submanifold

Let $X$ be a smooth $n$-manifold, with $n\ge 3$, such that $\pi_1(X)=\left<a_1,\ldots,a_m\right>$ (free group over $m$ elements) and suppose that there is an embedding: $$S^1\times ...
1
vote
0answers
18 views

Does weak Hausdorffication preserve equalizers and finite products?

There is a "weak Hausdorffication"-functor $wh$ from the category of compactly generated spaces (CG) to the category of compactly generated weak Hausdorff spaces (CGWH) given by quotienting out the ...
6
votes
1answer
113 views

Is $(\#^k \Bbb{RP}^2) \times I$ an $\mathbb{RP}^2$-irreducible 3-manifold?

Consider $S$ a surface homeomorphic to a connected sum of $n$ projective planes, $n \geq 2$. Can there be a two sided projective plane embedded in $[-\epsilon,\epsilon]\times S$?
0
votes
0answers
50 views

An algebra problem from spectral sequence [duplicate]

Recently, I am reading the article "You Could Have Invented Spectral Sequences" by Timothy Y. Chow. Link: http://www-math.mit.edu/~tchow/spectral.pdf In page 17, he used the following splitting which ...
0
votes
1answer
55 views

Triangulation of Torus in three different ways, but two of them are wrong.

In my geometry notes the writer states that the following two, are not triangulations for the torus: On the contrary this is a good triangulation: I tried to wrap a piece of paper in order the ...
2
votes
1answer
81 views

Why do Wikipedia and nLab seem to give completely incompatible definitions of the term “simplex”?

nLab seems to define that a simplex is an inhabited finite totally-ordered set. Wikipedia seems to define that simplex is a subset of real affine space satisfying certain conditions. Q. What's ...
0
votes
1answer
22 views

first homolgy group of a disk with $n$ holes

Let $D^2$ be a 2-dim disk with $n$ holes, i.e $D^2\setminus(S^0 * D^2)^n$. Then is it true that the first homology group of this space is $\mathbb{Z}^n$.
0
votes
1answer
35 views

Why this cocycle $\mathrm{char} (h)$ is not a coboundary?

Maybe this is a stupid question and I'm missing something very trivial. Let $X$ be a smooth manifold, $$h \colon Z_{k-1} (X, \mathbb{Z}) \rightarrow \mathbb{R}/\mathbb{Z}$$ an abelian group morphism ...
0
votes
0answers
40 views

Quotient homeomorphic to product

$X_1, X_2$ are topological spaces and $G_1,G_2$ are groups acting freely and properly discontinuously on these spaces by homeomorphisms. (Means that for every $g\in G$ the map $(g,x) \mapsto g(x)$ is ...
5
votes
2answers
91 views

Classify Open Sets in $\mathbb R^2$

In $\mathbb R$, we know that connected open set is $(0,1)$ under homeomorphism. I am wondering what is the situation in $\mathbb R^2$. From $\mathbb R^2-\text{pt}\simeq S^1$, we will have two open ...
3
votes
0answers
25 views

Homology of manifolds using submanifolds [duplicate]

Motivation I have learned algebraic topology. In simplical homology, we define $C_k(X)$ as an abelian group freely generated by $k$-dimensional skeleton $X^{(k)}$, and boundary operator $\partial_k$ ...
6
votes
1answer
93 views

Nontrivial cup product realized in $\Bbb R^4$

Let $A$ be a closed subspace $A$ of $[0,1]^4$---let's say, a subcomplex of some triangulation of the cube. I would like to show that the cup product $H^2(A)\times H^2(A)\to H^4(A)$ is trivial (or at ...
2
votes
1answer
82 views

Maps from Sum of Projective Planes to Circle

this is a problem from Lee's Topological Manifolds, 11-21. It asks the following: What compact, connected surfaces $M$ admit a non-nullhomotopic map to the circle? So, we use the classification of ...
1
vote
1answer
28 views

compact oriented $4k$-manifold-> euler characteristic is congruent to the signature mod $2$

Let $M$ be a compact oriented $4k$-manifold, $\chi_M$ the euler characteristic of $M$ and $sig(M)$ the signature of $M$. Why is $\chi_M \equiv sig(M)$ mod $2$? In the book " a concise course in ...
1
vote
1answer
21 views

Why is the sectional shape of a simply connected, oriented 4-manifold an isomorphism?

Let $M$ be a simply connected, compact and $\mathbb{Z}$-oriented 4-dimensional manifold. Let $\mu\in H_4(M;\mathbb{Z})$ be a fundamental class of $M$ (here is $H_4(M;\mathbb{Z})$ the singular homology ...
2
votes
1answer
24 views

Does weak Hausdorffication preserve inclusions?

There is a "weak Hausdorffication"-functor $wh$ from the category of compactly generated spaces (CG) to the category of compactly generated weak Hausdorff spaces (CGWH) given by quotienting out the ...
2
votes
1answer
67 views

Homology of sphere-complements

I have to solve the following questions: "For a subset $X \subset S^n$ determine the homology group $H_i(S^n - X)$, where (a) $X \cong S^l \vee S^k$ (b) $X \cong S^l \sqcup S^k$ (disjoint union) " ...
1
vote
2answers
55 views

$f(z) = z^3 + 2z + 7$. Calculate $f_* : H_2\to H_2$.

Let $f(z) = z^3 + 2z + 7$. $f: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$, with $f(\infty) = \infty$. Calculate $f_* : H_2(\hat{\mathbb{C}}, \mathbb{Z}) \to H_2(\hat{\mathbb{C}}, \mathbb{Z})$. What I ...
2
votes
1answer
36 views

Fundamental Group of Solid Octagon with Labelling Scheme

I am studying for the qualifying exams I am taking last week and am struggling with the following problem: Here's what I've tried. Say we call the left side $X$ and the right side $X'$. Pick ...
3
votes
1answer
59 views

Schubert decomposition of a Grassmannian

I'm going through Sheldon Katz's Enumerative Geometry and String Theory, and a few things regarding the Grassmannian $G(2,4)$ (lines in projective space) are bothering me: How can I compute the ...
2
votes
2answers
68 views

Openness condition in Seifert-van Kampen Theorem

I'm starting to learn some algebraic topology now, and came across the "classical" version of the Seifert-van Kampen theorem, whose statement is given in Theorem $4.5.2,$ on page $69$ here. If $X ...
3
votes
2answers
52 views

Reference for a vector space lemma of Hopf?

I've been told that the following is due to Hopf. Let $A, B, C$ be complex vector spaces. Given any linear map $$v:A\otimes B \rightarrow C,$$ where $A, B, C$ are complex vector spaces and $v$ is ...
3
votes
0answers
41 views

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 ...
0
votes
2answers
54 views

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 ...
11
votes
0answers
127 views

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 ...
3
votes
1answer
106 views

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: ...
1
vote
2answers
44 views

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 ...
1
vote
3answers
57 views

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 ...
0
votes
0answers
28 views

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 ...
-1
votes
1answer
63 views

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 $f$ ...
5
votes
1answer
65 views

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 ...
0
votes
1answer
43 views

Covering space action

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 ...
3
votes
1answer
55 views

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 ...