# Tagged Questions

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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### Applications of Topological Complexity of configuration space

I'm starting to work on Topological Complexity of configuration spaces . Articles say that it has applications in robotic and control theory . My questions are : 1) How Topological complexity can help ...
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In his article Construction of universal bundles. II (1956), John Milnor defines the strong topology in a join of spaces, but his definition is By a strong topology in $A_1\circ A_2\circ \dots \... 0answers 39 views ### Is there a general way to tell whether two topological spaces are homeomorphic? We know that if two topological spaces$X$and$Y$are homeomorphic, then they have the same fundamental groups, and the same homology. In other words, we have functors $$\pi_1 : \mathsf{Top} \to \... 0answers 63 views ### example of toric varieties with nontrivial first cohomology group If I remember correctly, when a toric variety is smooth or simplical (the moment polytope is simplicial rational), then there is no odd dimensional cohomology group and the dimension of the even ... 1answer 33 views ### Topological Join of Unit Balls I have seen that apparently one has for spheres that S^n*S^m=S^{n+m+1}. Is there a similar result for unit balls? Thank you. 2answers 138 views ### Lifting homeomorphisms covering Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf I don't understand the proof of Lemma 5.1. Notation: T_{0,0} is the 2-sphere, T_{... 1answer 29 views ### Construct smooth mapping f: B^{n + 1} \to S^n with two singularities at which f has degree +/- 1. I'm currently working through a paper by Pjotr Hajlasz who wants to show that For smooth manifolds M,N, if \pi_{[p]}(N) \neq 0 and 1 \leq p < n = \dim M, then the smooth mappings C^\... 1answer 72 views ### Show that the homology induced homomorphism f_*:H_3(RP^3)\rightarrow H_3(S^2\times S^1) is a zero map. Let f:\mathbb{RP}^3\rightarrow S^2\times S^1 be a continuous map. Prove that induced map f_*:H_3(\mathbb{RP}^3)\rightarrow H_3(S^2\times S^1) is a zero map. I found that the third homology of ... 1answer 30 views ### Interpretation of points in covering spaces as homotopy classes of paths [on hold] If p:\widetilde{X} \to X is a covering map, y \in \widetilde{X} determines a homotopy class of paths in X joining the base point x_0 to the point p(y). But a homotopy class of paths in X ... 2answers 65 views ### Fundamental group contains \mathbb{R} or \mathbb{Q} Is there any topological space whose fundamental group contains \mathbb{Q} or \mathbb{R}? In case of (singular) homology or cohomology, we can change its coefficients to any abelian groups (with ... 1answer 61 views ### For n\geq 2, any continuous map f:\mathbb{C}P^n\rightarrow S^2 induces the zero map on H_2(*) I am working through an old qualifying exam from another university. My course did not cover as much material as what is on this test (e.g. we did not cover cohomology). So I am just working through ... 1answer 87 views ### Why the attachment to simplices in (co)homology? I've been thinking a bit about why we define the singular homology and cohomology groups with simplices rather than, say, cubes, and it seems to me that the elementary aspects of the theory would all ... 1answer 62 views ### Counter-example for \tilde{H} (X/A) \cong H (X, A)? Yo! I was not able to find a counter-example to$$\tilde{H} (X/A) \cong H (X, A)$$. It's a well known fact that for cofibrations A \hookrightarrow X (or more generally whenever A is a deformation ... 1answer 59 views ### If two maps are homotopic, are the images homotopy equivalent? My question is; If two continuous maps f,g:X\rightarrow Y are homotopic, are the images Im(f),Im(g) homotopy equivalent? Clearly, the converse is false. If it is false, is there any condition ... 2answers 168 views ### Broken line is NOT diffeomorphic to the real line This is from Bredon's Topology and Geometry, page 71. This comes right after the very definition of differentiable manifold, so I think no use of tangent space or 'differential' is permitted. (Bredon ... 1answer 20 views ### Question About Covering Space Classification Theorem I'm a bit confused by Hatchers choice of words here. He says "The main classification theorem for covering spaces says that by associating the subgroup p_{*}(\pi_{1}(\tilde{X},\tilde{x_{0}})) we ... 0answers 25 views ### A doubt in Whitehead's proof about cohomology with local coefficients [on hold] In the proof of Theorem 4.9 says that p^*:H^n(X_n,X_{n-1};G|X_n) \to Hom(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0) has image Hom^{\pi}(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0). ... 1answer 28 views ### Map inducing zero on first cohomology is nullhomotopic (plus assumptions on fundamental group and universal cover) Currently I am studying for a topology exam next week and came across an exercise where I could need some hints (cf. here): Let X be a path-connected space with \pi := \pi_1(X,*) abelian and ... 0answers 18 views ### Explicit isomorphism H (E, E^0) \cong H_{cv} (E) Yo! I've been trying to understand better why$$H (D, D-\{x\}) \cong H_c (D)$$for a disk D and, more generally, why$$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ... 0answers 95 views ### On comparing two different notions of compactly generated space I have encountered, in different circumstances, the following two slightly different categories: The full category of \mathsf{Top} consisting of all objects that are: a) topological spaces ... 0answers 15 views ### Product and Join of G-CW-Complexes Given a topological group G and two G-CW-Complexes X and Y I want to understand the natural CW-structure on X\times Y and X*Y. I understand that the concepts are very similar, so I want to ... 1answer 36 views ### If 2 loops with equal base points are homotopic, must they be homotopic relative to the base point? Let X be a topological space and \mathbb {S}^1 be the set of complex numbers with magnitude 1 equipped with the inherited topology from \mathbb {C} . If we have 2 loops f,g:\mathbb {S}^1\... 1answer 436 views ### covering space of 2-genus surface I'm trying to build 2:1 covering space for 2- genus surface by 3-genus surface. I can see that if I take a cut of 3-genus surface in the middle (along the mid hole) I get 2 surfaces each one ... 1answer 31 views ### Difficulties with the description of p* in the serre spectral sequence(Bullet 7 of Mosher Tangora Page 76): For the first difficulty, let E^r be the rth page of a first quadrant spectral sequence with elements E^r_{p,q} , where p is the filtering degree. Difficulty 1: On bullet 7 of Mosher and ... 2answers 44 views ### Principal bundle as homotopy fiber universally self-trivializes In this MO answer, I was told the definition of principal bundle as a homotopy fiber of its classifying map precisely says that it's the universal bundle which trivializes itself. However, I'm having ... 2answers 23 views ### A relation between interior and closed sets A topological space X is said to be completely regular provided that it is a Hausdorff space such that, whenever F is a closed set and x is a point in its complement, there exists a function f\... 0answers 48 views ### Why all differentials are 0 for Serre Spectral Sequence of trivial fibration? Consider the fibration F \hookrightarrow F \times B \to B. I understand that if I take kunneth's theorem for granted that the group extensions F_{n-i,i} \to F_{n-i+1,i-1} \to F_{n-i+1,i-1}/F_{n-i,... 1answer 47 views ### Fixed-point free map of the 2-sphere which has order 4 The antipodal involution of \mathbb{S}^2 clearly has no fixed points. However I cannot think up an example of homeomorphism of order 4 which has no fixed points. Could you help me? 1answer 22 views ### Steenrod Operations an algebraic Approach Assume that q=p^{r}, where p is a prime either 2 or odd and \mathbb{F}_{q} is a Galois field and V a finite dimensional \mathbb{F}_{q}-vector space. Then due to Larry Smith in this http://... 2answers 57 views ### How to show that \pi_1(\mathbb{RP}^2) = \mathbb{Z}_2? How to show that \pi_1(\mathbb{RP}^2) = \mathbb{Z}_2? In general is well known that \pi_1(\mathbb{RP}^n) = \mathbb{Z}_2, ~ n \ge 2. But how to show this assertion? I have a few knowledge about ... 0answers 44 views ### Prove that the fundamental group of P^2 the real projective plane is the group with two elements Presumably the group will just contain e and another g. My idea is to use the proof that \pi_1(S^1)=\mathbb{Z} where the covering map from S^2 to P^2 maps to the line in \mathbb{R}^3 ... 2answers 62 views ### Topology of complex projective plane It is well known there are two ways to construct topology of \mathbb{C}P^n: quotient space of S^{2n+1} by identifying x with \lambda x, where \lambda is complex nonzero constant. According ... 1answer 39 views ### Can we lift paths of a Lie group quotient G\to G/H? Question: Let G be a Lie group and H\subseteq G a closed normal subgroup. Let$$\pi:G\to G/H$$be the quotient map. If \gamma:[0,1]\to G/H is a smooth path, can we find a smooth path \tilde{... 0answers 28 views ### Universal Abelian Covering Space of genus two surface [closed] Let M be a surface M, i am concerned with Abelian covers. These are the covering spaces for which the deck group is Abelian. The largest such cover corresponds to the commutator subgroup of the ... 0answers 24 views ### The action of \pi_1(BK) \curvearrowright H_*(BG) for the fibration BG \hookrightarrow BH \to BK **Note:I was extremely confused when I wrote this post. Please see the linked one. I left this one as it is, because what I understand now is so radically different then what I wrote below ** Let ... 0answers 21 views ### basepoint problem: Is there an action of \pi_1(B) on \pi_1(F) for F path connected I am doing this to try to figure out The action of \pi_1(BK) \curvearrowright H_*(BG) for the fibration BG \hookrightarrow BH \to BK . Let the fibration F \hookrightarrow E \xrightarrow{p} B be ... 1answer 36 views ### Proof of the cellular boundary formula I'm trying to understand the proof in Hatcher (p. 141) of the cellular boundary formula. Now there's one thing that Hatcher does several times in his book and that I don't understand very well: he ... 0answers 25 views ### If p: E \rightarrow X is a covering map with E connected and |p^{-1}(x_{0})|=k for some x_{o} then |p^{-1}(x)|=k for all x \in E. Prove that if p:E \rightarrow X is a covering map with E connected and p^{-1}(x_{0}) has k elements for some x_{0} \in X, then p^{-1}(x) has k elements for every x \in X. Is my proof ... 1answer 73 views ### Action of \mathbb{Z}/3\mathbb{Z} on P^{1} I am reading from the book Topics in Galois theory by Serre. I have the following question , take G=\mathbb{Z}/3\mathbb{Z}. The group G acts on P^1 by$$\sigma x\;=\;1/(1-x)$$where$\sigma$... 1answer 31 views ### verifying homeomorphism of orbit space and suggestions for further study Define an action of$\mathbb{Z}_2$on$S^1$by$(0,z)\mapsto z$and$(1,z)\mapsto \bar{z}$. An orbit of$z$is then the set$\{z,\bar{z}\}$. I claim the orbit space$S^1/\mathbb{Z}_2$is homemorphic ... 1answer 99 views ### Is the nth homotopy group isomorphic to$[T^n, X]$Following Spanier's book on algebraic topology chapter$1$, section$6$about suspensions, I'm wondering about the following questions: 1) We know that$S^n$is an$H$-cogroup for all$n\geq1$... 1answer 166 views ### When can we recover a manifold when we attach a$2n$-cell to$S^n$? I have a question related to this one. In my answer I was going to try and say something about the possible manifolds that might arise in this way, i.e. as mapping cones of elements of$\pi_{2n-1}(S^...
In Hatcher's book we find, when computing the boundary maps of cellular homology, Cellular Boundary Formula: $d_n(e^{n}_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_{\beta}$ where $d_{\alpha\beta}$ is ...