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Questions tagged [algebraic-topology]

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

3
votes
1answer
33 views

Are products of spheres $\prod_i\Bbb S^{n_i}$ different?

If $(n_i)_{i\leq k}$ and $(n_i^\prime)_{i\leq k^\prime}$ are increasing (not necessarily strictly) sequences of non-zero integers, do we have the following? $$\prod_{i\leq k}\Bbb S^{n_i}\simeq \...
1
vote
0answers
8 views

If $\{p_0,\dots,p_m\}$ is affine independent with barycentre $b$, then $\{b,p_0,\dots,\hat{p}_i,\dots,p_m\}$ is affine independent for each $i$.

My question is how to solve this problem. I feel like I am missing something obvious here. (This is Exercise 2.9 in Rotman's Introduction to Algebraic Topology.) If $\{p_0,\dots,p_m\} \subset \...
9
votes
2answers
78 views

Homotopy equivalence upper triangular matrices and torus

In an old algebraic topology exam, I came across this question. Let $G$ be the set of invertible upper triangular matrices in $\mathbf{C}^{2\times 2}$, as a topological subspace of $\mathbf{C}^3\...
2
votes
1answer
36 views

A free action is not necessarily a covering space action: counterexample

In p.73 of Hatcher's algebraic topology, he gives an example of a free action which is not a covering space action. An example is the action of $\mathbb{Z}$ on $S^1$ in which a generator of $\...
4
votes
1answer
31 views

Issue with Coequalizer Definition of the Horn of a Simplex

I am having trouble understanding the maps in this coequalizer defining the k-horn (from page 9 of Goerss and Jardine's Simplicial Homotopy Theory). It is defining them using the ith, jth inclusion ...
3
votes
1answer
48 views

Is the Kunneth formula for de Rham cohomology true on the cochain level

The kunneth formula gives that $H^k(X \times Y) = \bigoplus_{i+j = k} H^{i}(X) \otimes H^{j}(Y)$, where $X$ and $Y$ are both manifolds. I wonder whether this is true on the cochain level. More ...
2
votes
0answers
30 views

What's wrong with this argument that the Atiyah-Hirzebruch spectral sequence always degenerates?

Let $E$ be a spectrum and let $X$ be a space or connective spectrum. Then the cohomological Atiyah-Hirzebruch spectral sequence is of the form: $H^s(X,E^t) \Rightarrow E^{s+t}(X)$ This is a half-...
1
vote
1answer
49 views

$\mathbb{S}^n$ without two points

In "An Introduction to Algebraic Topology" of Rotman, Exercise 1.31 asks to show that the equator $\mathbb{S}^{n-1}$ is a deformation retract of $\mathbb{S}^n\setminus\{a, b\}$. I thought that if ...
1
vote
1answer
46 views

Hatcher Theorem 3.44

Try to prove the commutativity of the diagram in Hatcher Theorem 3.44. Sadly, I am stuck on the commutativity of the following diagram. Let $M$ be a closed orientable manifold of dimension $d$, i.e $...
1
vote
1answer
38 views

Connection between fundamental group of circle and division by zero

Going through some of the applications of the proof that $\pi_{1}(S^{1}) \cong \mathbb{Z}$ in Hatcher, I have noticed that almost all of these arguments rely on a contradiction whose assumption ...
3
votes
2answers
24 views

Question on calculating $H_n(S^1 \times S^1,A)$ where $A$ is a finite set of points

So, around $H_1(S^1 \times S^1,A)$, our long exact sequence looks like: $0 \rightarrow H_1(S^1 \times S^1) \rightarrow^h H_1(S^1 \times S^1,A) \rightarrow^g H_0(A) \rightarrow^f H_0(S^1 \times S^1) \...
3
votes
0answers
31 views

Problem with Serre Fibration between Classifying spaces

My problem is from Bruno Harris paper "Bott Periodicity via Simplicial Spaces" Let $\mathcal{G}$ be the category, $Ob(\mathcal{G})=Gr= \coprod_n Gr_n(\mathbb{C^\infty})$ with only identity morphisms. ...
2
votes
2answers
28 views

Show that two embeddings of $M$ into its product are not homotopic

Assume that $M$ is a compact smooth manifold with positive dimension. We have two ways of embedding $M$ into its product with itself. Way I: $ i_1(m) = (m, m)$ and Way II: $i_2(m) = (a, m)$, where $a \...
5
votes
1answer
45 views

When do elements of $\operatorname{Hom}(G,G)$ correspond to invertible self maps of $K(G,n)$?

Suppose we pick a natural isomorphism between $H^n(-;G)$ and $\langle -, K(G,n)\rangle$, when does an element of $H^n(K(G,n),G)=\operatorname{Hom}(G,G)$ correspond to a self map of $K(G,n)$ that has a ...
2
votes
0answers
21 views

K-theory of classifying spaces

Can someone help me calculate the following groups in $ K $-theory 1) $ KU^0 (B\mathbb{S}^1) $ 2) $ KU^0 (\mathbb{RP}^\infty) $ where $ B \mathbb{S}^1$ is the classifying space of $ \mathbb{S}^1 $ ...
3
votes
1answer
62 views

Computing $[\mathbb{R}P^2, S^k]$

I am trying to compute $[\mathbb{R}P^2,S^k]$ for $k\geq 0$, via the cofiber sequence associated to $f:S^1\to S^1$ given by $z\mapsto z^2$, where we get the mapping cone $C_f \cong \mathbb{R}P^2$. The ...
1
vote
1answer
28 views

Cohomology of classifying space

I would like to know if anyone knows how to calculate the cohomology of the following spaces, especially in the case of classifying spaces: 1) $ H^\ast (BSU(2), \mathbb{Z}) $ 2) $ H^\ast (BO(3), \...
2
votes
1answer
42 views

Why are local diffeomorphisms between spheres are actually diffeomorphism

If $f: S^n \to S^n$ be a local diffeomorphism and $n > 2$, then $f$ is a global diffeomorphism. I do not know why this should be true. I tried thinking of the $n = 1$ case, $z^2$ is a local ...
6
votes
2answers
76 views

Covering map from sphere with six points removed to doubly-punctured complex plane

$X$ is $S^2\subset \mathbf{R}^3$ with its intersection points with the coordinate axes removed. Show that the following map is a covering map. $$\begin{align*}p:X&\longrightarrow \mathbf{C}-\...
0
votes
1answer
39 views

Local systems as solution of PDE

I want to understand the correspondence between locally constant sheaves and vector bundles with flat connection on a manifold $X$. Given a local system $\mathcal L$, it is clear how to define a ...
1
vote
1answer
20 views

quotient map of locally compact spaces [on hold]

Suppose $X,Y$ are non-compact locally comapct Hausdorff spaces,suppose we have a quotient map $Q:X\to Y$,can we conclude that $Q$ is proper?
-2
votes
0answers
35 views

Fulton, Example 3.2.8 [on hold]

Example 3.2.8 Let $X$ and $Y$ schemes, $p$ and $q$ the projections from $X \times Y$ to $X$ and $Y$, $E$ and $F$ vector bundles on $X$ and $Y$, $\alpha \in A_{*}\alpha$, $\beta \in A_{*}Y$. Then \...
1
vote
2answers
37 views

Hatcher's proof of proposition (c) for covering spaces

I am currently going through the proof of Hatcher's Algebraic Topology, and I am having some difficulty regarding one of his assertions in the section of the proof given below: I am unable, in ...
4
votes
0answers
30 views

What is the $S^1$-equivariant cup product on $S^2$?

Consider the sphere $S^2 = \mathbb{CP}^1$ with the $S^1 = \{ \tau \in \mathbb{C} \mid |\tau| = 1 \}$ action given by $$ \tau \cdot [z_1, z_2] = [\tau ^ k \cdot z_1, z_2] $$ The corresponding $S^1$-...
3
votes
0answers
32 views

Wu formula for $\mathbb{Z}_N$ classes

Let $w_1\in H^1(M, \mathbb{Z}_2)$ be the Stiefel-Whitney class of the tangent bundle of d-dimensional manifold $M$, and $x_{d-1}\in H^{d-1}(M, \mathbb{Z}_2)$. Wu formula tells us $$Sq^1 x_{d-1}= u_1\...
1
vote
1answer
33 views

Singular Homology of Real Projective Spaces

I'm following an induction argument to calculate the singular homology of $\mathbb{R}\mathbb{P}^n$ with coefficients in $\mathbb{Z}$. We decompose $\mathbb{R}\mathbb{P}^n$ into $U := \{[x_0 : \cdots : ...
0
votes
0answers
43 views

What is $P^2$ with a disk removed

Recall that $P^2$ can be obtained by attaching a 2 cell to $P^1$. Denote this 2 cell by $B_2$. Now remove an open disk from $B_2$, according to this post it should be the Mobius strip. However it ...
1
vote
1answer
31 views

Homology of the complement of a compact subset inside ball

Let $A$ be a abelian group and let $\mathbb{B}^m=\{|x|<1\}$ and let $L\subseteq \mathbb R^m$ be compact such that $K:=L\cap \mathbb B^m\ne\emptyset$ is connected. Clearly we have an iso $$H_{m-1}(\...
2
votes
1answer
88 views

Relationship between homotopy pushout and ordinary pushout

I'm trying to understand the homotopy pushouts and currently looking at the homotopy cofiber. For two maps $f \colon C \to A$ and $g \colon C \to B$ we defined the homotopy pushout to be the regular ...
2
votes
2answers
32 views

Why $f^{'}_{*}$ group homomorphism exist in 'only if' part of the lifting criterion proof?

In Hatcher, the lifting criterion states (Prop 1.33): Suppose given a covering space $p: (X^{'},x^{'}) \rightarrow (X,x_0)$ and a map $f: (Y,y_0) \rightarrow (X,x_0)$ with $Y$ path-connected and ...
1
vote
0answers
30 views

calculating the De Rham cohomology of $\mathbb RP^2$ using Mayer Vietoris

View $\mathbb RP^2$ as $\mathbb RP^1$ attached with a 2-dimensional open unit ball $e^2$. Let $U$ be an open neighborhood of $\mathbb RP^1$ and $V$ be $e^2$. Then $U \cap V$ deformation retracts onto ...
0
votes
0answers
54 views

Proof of existence of spectral sequence.

I'm trying to understand the proof of existence of spectral sequence from Ch5 of Allen Hatcher's Spectral Sequence notes. While constructing the iso $\tilde{\Phi_*}$ in the diagram it's written that: ...
0
votes
0answers
33 views

Thom class in homology for defining orientability of vector bundles

A rank $r$ real vector bundle $p : E \to B$ is said to be orientable if there is a Thom class $\tau \in H^r (D(E),\partial D(E) ; \mathbb{F})$ (where $D(E)$ is a unit disk bundle and $\mathbb{F}$ a ...
0
votes
1answer
37 views

Confusion about topology on CW complex: weak or final?

The topology of the CW complex is defined to be the weak topology: given the sequence of inclusions of the skeleta $X_0 \subseteq X_1 \subseteq_ \cdots$ a subset $A \subseteq X = \cup X_i$ is open iff ...
8
votes
1answer
57 views

Corestricting a weak homotopy equivalance

Let $X$ and $Y$ be topological spaces. Let $f: X \to Y$ be a continuous map. Recall that $f$ is a weak homotopy equivalence iff $f$ induces group isomorphisms on the homotopy groups, i.e.: $$\...
0
votes
0answers
113 views

on cross section of fiber bundle

Let $T$ be a topological group acting continuously from the left on a compact Hausdorff space $X$, and let $G$ be the circle group acting freely and continuously from the right on $X$ such that $t(xg)=...
1
vote
1answer
45 views

Show that the Riemann sphere complement the unit disc is a hyperbolic Riemann surface

I am trying to understand what $\hat{\mathbb{C}} \backslash \bar{\mathbb{D}}$ in the area of Riemann surfaces. $\hat{\mathbb{C}}$ is the Riemann sphere and $\bar {\mathbb{D}}$ is the closure of the ...
1
vote
0answers
31 views

$\sin$ as covering map between Riemann surfaces [duplicate]

$\newcommand\C{\mathbb C} \newcommand\bs{\backslash}$There is a book on Riemann surfaces by Forster that I like very much. I am curious about the first exercise in chapter 1 section 4 on branched and ...
0
votes
0answers
43 views

K theory of integers [closed]

How can i calculate $K_2$ of $\mathbb{Z} $ i.e. $K_2 (\mathbb{Z})$
4
votes
1answer
84 views

Good list of exercises in Hatcher’s book algebraic topology

I’m doing self study of Hatcher’s book on algebraic topology. There are too many exercises after each section and try to solve all the exercises needs a lot of work and definitely not an efficient and ...
0
votes
1answer
15 views

Definition of Prism Operator and homotopy of chain complexes

I am reading Algebraic Topology by Allen Hatcher. For reference: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf On page 112, in the first sentence of the last paragraph, we find the following: The ...
3
votes
1answer
63 views

relative homology of manifold with boundary

I don't understand the following statement from Bredon Lemma 9.1. The statement is: For any connect, compact manifold $M^n$, Let $A$ be a component of $\partial M$, and let $B = \partial M - A$. ...
1
vote
1answer
69 views

Fundamental group of collapsed cube skeleton

I’m having trouble solving this exercise. Let $C=[0,1]^3$ be the standard cube in $\mathbb{R}^3$ And let $X$ be the edges. Now I should compute the fundamental group of $Y$ where $Y$ is ...
2
votes
0answers
29 views

Functoriality of parallel transport of a Hurewicz connection on a fiber bundle

Let $A\overset{\alpha}{\rightarrow}B$ be a Hurewicz fibration. Any Hurewicz connection defines parallel transport along curves in the base. In general, such parallel transport maps $\alpha^{-1}(b)\to \...
1
vote
0answers
24 views

Prove that there is no retraction from $X$ to the image of a loop $\gamma(t)$

Let $X = S^1 \times D^2$, pick $x_0 \in X$. Suppose $\gamma$ is an embedded loop based at $x_0$ which represents an $n \in \mathbb{Z} = \pi_1(X)$ with $n \neq \pm1$. Prove that there is no retraction ...
1
vote
1answer
13 views

Crushing a central dividing circle in $\Sigma_2$ to a point

Crushing a central separating circle in $\Sigma_2$ to a point So presentation of $\Sigma_2$ is $<x_1,y_1,x_2,y_2:[x_1,y_1][x_2,y_2]>$. I understand that crushing this separating circle to a ...
0
votes
1answer
28 views

Reference for homology of real projective space in a field.

I need a reference in a book for the computation of the homology of real projective space with coefficients in an arbitrary field. I do know how to do the computation, and I also found an online ...
3
votes
1answer
66 views

Homology Represents topological Subspaces

Consider $X:= \mathbb{PC}^n $ the projective space. It is well known that the integral homology of $X$ is $H_i(X, \mathbb{Z}) = \mathbb{Z}$ is given by: $0 \leq i \leq 2n$ even, and $H_i(X, \mathbb{Z}...
1
vote
1answer
67 views

Show that $\bigoplus_{i \text{ even}}C_i=\bigoplus_{i\text{ odd}}C_i$

Let $C_*$ be a chain complex such that each $C_i$ is a torsion-free, finite-range abelian group with $C_i=0$ for all $i<0$. Suppose that $C_i=0$ for all $i$ is sufficiently large and that for all $...
1
vote
1answer
52 views

ext sheaf and cohomology

Let $\mathcal{F}$ be a sheaf on $\mathbb{P}^{3}$ with $\mbox{dim}(\mathcal{F}) = 0$. It's true that cohomology $H^{i}(\mathbb{P}^{3}, ext^{3}(\mathcal{F}, \mathcal{O}_{\mathbb{P}^{3}})) = 0$ for $i = ...