All questions with tag [math: algebraic-topology]


Fundamental group of $\mathbb{R}^{3}\setminus \{ \mbox{2 linked circles }\}$

Calculate the fundamental group of the complement in $\mathbb{R}^3$ of $$\{ (x,y,z) \ | \ y = 0 , \ x^{2} + z^{2} = 1\} \cup \{ (x,y,z) \ | \ z = 0 , \ (x-1)^{2} + y^{2} = 1\}.$$ Note: this space is $\mathbb{R}^{3}\setminus \{ \mbox{2 linked circles }\}$.
2022-07-25 20:47:21

cohomology group of $SO(n)$

I am computing the Alexander-Spanier cohomology $H^i(SO(n),\mathbb{Z})$. I embedded $SO(n)$ into $R^{n^2}$. Since the embedding $i$ is a monomorphism, the induced group homomorphism $i^*$ is an epimorphism. Since $R^{n^2}$ is homotopic to a point, $H^i(R^{n^2})=0 , \forall i \in \mathbb{Z}^+$. That gives us $H^i(SO(n))=0, \forall i \in \mathbb{...
2022-07-25 20:46:55

Are fibers of a fiber bundle the same as fibers of a covering space?

I was asking yourself exists any kind of distinction in between them? As all fibers are fiber packages, so undoubtedly the fibers coincide. Yet, after that could not there be some unique feature of a fiber of fiber package that isn't in covering rooms. As Hatcher does not specify fiber when he is defining what a fiber package is.
2022-07-25 20:46:44

Path lifting theorem

I'm attempting to generalise this theory. Yet, was asking yourself in the evidence offered below and also in a similar way in Hatchers. Can you change the $S^{1}$ with a basic room X. As it appears to not be that vital in the evidence. So if you pick the parts of a basic $X$ can not you make the reduction? So is $S^{1}$ actually that vital fo...
2022-07-25 20:42:31

orientability of the möbius strip using homology

I read in Hatcher is "Algebraic topology" publication concerning orientability of topological maifolds making use of homology. currently I would love to recognize just how one can use this to show that the möbius strip is not orientable? i have no suggestion.
2022-07-25 20:18:46

Pullback of a locally constant sheaf by a function whose domain is simply connected

Let $\mathcal{A}$ be a locally constant sheaf on a topological space $X$ and let $\sigma:\Delta_p\to X$ denote a singular $p$-simplex. Writing the pullback of $\mathcal A$ by $\sigma$ as $\sigma^\ast(\mathcal A)$, Bredon's book on sheaf theory (page 26 in the second edition) says: Since $\mathcal{A}$ is locally constant and $\Delta_p$ is simply...
2022-07-25 20:18:35

Showing higher homotopy groups of $S^1$ are trivial

I'm attempting to confirm $\pi_{i} (S^1) \cong 0$ if $i>1$. Is this proper. You have a brief exact sequence, $\mathbb{Z} \rightarrow \mathbb{R} \rightarrow S^1$ (from the fiber package of the covering room), after that you reason from some magic I do not recognize you can do this $\pi_{i}(\mathbb{Z}) \rightarrow \pi_{i}(\mathbb{R}) \rig...
2022-07-25 20:16:00

Confused about Hypercohomology terminology and meaning

check this: Given a sheaf complex $F^\bullet$, let's say I want to compute the hypercohomology of this complex, if we consider the bicomplex of sheaves $C^\bullet(F^\bullet) = (C^p(F^q))\quad (p,q\in\mathbb{Z})$, where $C^\bullet(F^q)$ is the Godement resolution of the sheaf $F^q$. The hypercohomology of $F^\bullet$ is the cohomology of the comp...
2022-07-25 20:06:49

Question on covering spaces

Let $q:X\to Y$ and $r:Y \to Z$ be covering maps;let $p=r\circ q$. Show that if $r^{-1}(z)$ is finite for each $z\in Z$, then $p$ is a covering space. I'm confused that $p=r\circ q$ is obvious surjective and each open set $U\subset Z$, $r^{-1}(U)=\cup_\alpha V_\alpha\subset Y$,and each $q^{-1}(V_\alpha)=\cup_\beta W_\beta \subset Z$ and each $W_\...
2022-07-25 19:56:58

Calculating Homology Groups of $S^1\times X$

Given that $H_n(X)$ is free abelian, I'm trying to find the homology groups of $Y=S^1\times X$ using the Mayer-Vietoris theorem. My first attempt decomposed $Y$ as $A \cup B$ where $A=\{*\}\times X$ and $B = S^1 \setminus \{*\}\times X$. Then $B$ clearly homotopy equivalent to $A$ and the Mayer-Vietoris sequence gives that $H_n(Y)=H_n(X)\bigoplu...
2022-07-25 19:55:57

Double Coverings of the Double Torus

I'm trying to count all the double coverings of the double torus. I know that the fundamental group of the double torus is $$\pi_1(X)=\langle a,b,c,d;[a,b][c,d]\rangle $$ where $[a,b]=aba^{-1}b^{-1}$. I also know from the classification theorem for covering spaces that in fact it suffices to count subgroups of index 2. I'm not sure how to do th...
2022-07-25 19:43:03

Some fundamental groups

Here are some workouts of algebraic topology (they aren't research yet self - researching) a) Calculate the fondamental team of the subspace of $$\mathbb{R}^3 H = S^{2} \bigcup \{ \text{the 3 coordinate planes} \}$$ b) Prove that for every single $ 0 \le n \le m - 3 \ $, the corresponding in $\mathbb{R}^{m} \ $ of a $n$ - dimensional vector ...
2022-07-25 19:37:10

Prove every map from the projective plane to the circle is nullhomotopic

Prove that every continual map $f:P^2\to S^1$, where $P^2$ is the projective aircraft, is nullhomotopic. I assume I require to make use of the reality that $\pi_1(P^2) = \mathbb{Z}/2\mathbb{Z}$ and also covering room concept. The map $f$ generates a map $f_* : \pi_(P^2) = \mathbb{Z}/2\mathbb{Z} \to \pi_1(S^1) = \mathbb{Z}$, yet I do not see why...
2022-07-25 16:29:56

Vanishing of higher cohomology

If $M$ is a manifold of measurement $n$, does single cohomology $H^i(M, \mathbb{C})$ disappear when $i > n$? If $M$ is an algebraic selection over $\mathbb{C}$, outfitted with average geography, can one claim something concerning the disappearing of higer single cohomology?
2022-07-25 16:20:00

Can $\mathbb RP^2$ cover $\mathbb S^2$?

I know $\mathbb{S}^2$ is the universal cover of $\mathbb{R}P^2$, but can $\mathbb{R}P^2$ be a covering space (at all) of $\mathbb{S}^2$? Attempt at solution It's clear that for a umramified covering with degree $n$ between surfaces, the euler characteristic of the covering space must be $n$ times the Euler characteristic of the image space. $\ch...
2022-07-25 16:14:58

Freudenthal suspension theorem- Weak excision lemma I'm attempting to recognize this theory on web page 6. Apparantly you can make use of that to confirm the Freudenthal suspension theory for rounds. Nonetheless, I assume it is incorrect specifically $\pi_{i}(A,C) \rightarrow \pi_{i}(X,B)$, should not it be An as opposed to B in ...
2022-07-25 15:51:31

when is the kth homology group of a space isomorphic to its kth homotopy group?

I'm simply thinking of the partnership in between homology and also homotopy teams of a room. I recognize that homology is primarily an abelianization of the basic team (please remedy me if I'm incorrect). If any person can please claim a couple of words concerning my inquiry, I would certainly appreciate. What/where else could I review resolvin...
2022-07-25 15:51:24

Approach to Learning Co/Homology

I have actually determined to begin researching co/homology and also I'm attempting to exercise the most effective strategy to doing this. As I recognize the scenario, any kind of system that pleases the Eilenberg - Steenrod axioms certifies as a Homology Theory. Details instances of homology concept include: Simplical Homology Singular Homol...
2022-07-25 15:37:08

Cohomology ring of Grassmannians

I'm reading a paper called An Additive Basis for the Cohomology of Real Grassmannians, which begins by making the following claim (paraphrasing): Let $w=1+w_1+ \ldots + w_m$ be the total Stiefel-Whitney class of the canonical $m$-plane bundle over $G_m(\mathbb{R}^{m+n})$ and let $\bar{w}=1+\bar{w_1}+\ldots+ \bar{w_n}$ be its dual. Then $H^\ast ...
2022-07-25 15:35:45

what is the degree of $f :S^n \to S^n$ when $f$ has no fixed points?

Let $n \ge 1.$ and also allow $f: S^n \to S^n$ be continual self - map of the device $n$ - round. If $f$ has no set factors, what is the level of $f$, and also why? Many thanks beforehand.
2022-07-25 15:34:21