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

learn more… | top users | synonyms (1)

0
votes
0answers
45 views

Invertibility of suspension in spectra

I know that spectra are supposed to be designed so that suspension is invertible up to homotopy, but I'm having trouble articulating exactly why this is the case. If $E$ is a spectrum and $\Sigma E$ ...
0
votes
0answers
65 views

A Theorm on Covering spaces

I was reading a theorem,precisely Theorem 2.2.10,page 33 of this book below- ...
0
votes
0answers
32 views

Composition of boundary homomorphism from Hatcher

We have $\partial_n ( \sigma) = \sum_{i} (-1)^{i} \sigma [ v_1, \dots, \overset { \wedge} v_i, \dots, v_n]$ and we want to show that the composition $$ \delta_n(X) \overset{\partial_n} \to ...
0
votes
0answers
17 views

stable splittings of projective space

On Hatcher's book Algebraic Topology, page 468 Prop. 4I.3, For prime number $p$, can we decompose $\mathbb{C}P^\infty$ in a similar way?
0
votes
0answers
28 views

Making a homotopy equivalence out of a pushout map involving $A \cup B = X$

Given the topological space $X$ with subspaces $A$, $B$ so that $A \cup B = X$ and the maps in the "square" of the following diagram ($i_1$, $f$, $g$, $h$) forming the pushout $Y$: I added rest of ...
0
votes
0answers
25 views

dual hopf algebras

Let $X$ be an H-space with product $\mu$. Let diagonal map $\Delta: x\mapsto (x,x)$. Let $F$ be a field. (1). Then by Kunneth formula, $H_*(X\times X;F)=H_*(X;F)\otimes H_*(X;F)$. (2). Hence $$ ...
0
votes
0answers
24 views

trivialization of a bundle

suppose we have a $D^2$-bundle $X$ over a surface with boundary $F$. It is said that we can always trivialize the bundle: X is diffeomorphic to $F\times D^2$ but I do not see why this is true.
0
votes
0answers
33 views

Recovering flow values given total values

I have the following problem which I am failing to put into a tractable Mathematical minimization problem. We are observing some flows. A flow can start at any month in a year and end in any month ...
0
votes
0answers
42 views

The generator of compact cohomology of punctured plane

Can anyone give detailed derivation of the generator of compact cohomology of $H^1_c (R^2-\{0\}$). (It is homotopic to a circle so it is isomorphic to $R$ but I want computation of its generator, ...
0
votes
0answers
224 views

The Complex projective space is homeomorphic to the n-sphere

Ok I have been asked to give as detailed a proof as I can for the following question. Prove that $ \mathbb C\mathbb P^n $ is homeomorphic to $ S^{2n+1} /\sim. $ where for $ z,w \in S^{2n+1} \subset ...
0
votes
0answers
41 views

Free action of symmetric group

What type of compact manifolds, can be acted freely by symmetric group $S_{m}$ for some $m>2$? Is there a compact manifold which can be act freely by all symmetric ...
0
votes
0answers
17 views

Homotopy classes of self-maps on L(2,1)#L(3,1)

How to classify continuous self maps on $L(2,1)\# L(3,1)$ up to homotopy? Here $L(2,1)$ and $L(3,1)$ are lens spaces. The reason for considering this manifold is that its fundamental group is ...
0
votes
0answers
50 views

abelianized fundamental groups.

I am trying to show that there is a canonical ismorphism between the abelianized fundamental groups, $\pi_1(X,p)_{ab}$ and $\pi_1(X,q)_{ab}$ of the path-connected space $X$. I know since $X$ is path ...
0
votes
0answers
34 views

classifying map of associated bundles

Let $G\leq GL(\mathbb{R}^n)$ be a group and $\xi$ be a principal $G$-bundle over a space $X$. Let $\eta=\xi[\mathbb{R}^n]$ be the associated vector bundle of $\xi$. Let $f_\xi: X\to BG$ be the ...
0
votes
0answers
39 views

strengthening Lesbegue Number Lemma

Let $F : I \times I \rightarrow X$ be a continuous map and U, V be two open covers of X. Then Lesbegue lemma says that there exist partitions of I which are $0=s(0)<...<1=s(m)$ and ...
0
votes
0answers
43 views

Twisted cohomology as sections of bundle of Eilenberg-Maclane spaces

Let $X$ a space, and $E$ an multiplicative cohomology theory represented by a ring spectrum $K$, i.e. $E^\bullet(X)=[X,K]$. Also let $A$ be a local system of abelian groups. Cohomology with local ...
0
votes
0answers
111 views

Geometric Realization of Finite Dimensional Abstract Simplicial Complex

I am learning the theory of complex. And there are two theorems presented by our teacher: Every abstract complex $K$ has its geometric realization. Every $n$-dimensional abstract complex $K$ has its ...
0
votes
0answers
59 views

Roots of monic complex polynomial lie on a circle of radius $R$.

A problem in my topology course asks to show that there exists a large enough $R$ such that $f(x)=z_n x^n + \dots + z_0$ has no roots on $ \mid z \mid =R$. I am not sure how to approach this problem ...
0
votes
0answers
56 views

Additivity for Relative Homology

If $(X_\alpha,A_\alpha)$ are disjoint topological pairs, is the following statement true? $$ \bigoplus_\alpha H_n(X_\alpha,A_\alpha)=H_n\left(\bigcup_\alpha X_\alpha,\bigcup_\alpha A_\alpha\right) $$ ...
0
votes
0answers
51 views

Identifying Objects with Polygons

I can't seem to find anything regarding how one identifies something like a torus with am oriented square. I would like to know the significance of: How does the rectangle depict the torus? Why are ...
0
votes
0answers
42 views

What is the “product rule” for the boundary map of a product of CW-complexes?

I'm reading some notes that compute the action of the boundary maps on the CW-complex $\mathbb{R}P^2\times\mathbb{R}P^2$. I know the chain maps for $RP^2$ itself are $d_0,d_1\equiv 0$, and $d_2=\cdot ...
0
votes
0answers
39 views

What does it mean to say a simplex has all its vertices distinct?

In Hatcher I'm trying to solve the problem: Show that the second barycentric subdivision of a $\Delta-complex$ is a simplicial complex. Namely, show that the first barycentric subdivision produces a ...
0
votes
0answers
40 views

$n$-connectivity of principal $G$-bundle

Page 202 of Switzer's Algebraic Topology: Let $\xi=(E,p,B)$ be a principal $G$-bundle such that $E$ is $n$-connected. Then for any pointed CW-complex $(X,x_0)$ of dimension at most $n$, the ...
0
votes
0answers
63 views

How do i prove that this is homeomorphic to Klein Bottle?

My professor is teaching quotient space recently and he gives very informal arguments (drawing diagrams,arrows and stuff). I think he wants students to get familiar with quotient spaces and visualize ...
0
votes
0answers
78 views

Algebraic K-theory: Categories of modules and their equivalences

I'm currently preparing a presentation on the second chapter, called "Categories of modules and their equivalences", in Algebraic K-theory by Hyman Bass. I have a VERY elementary understanding of ...
0
votes
0answers
36 views

A question from May's notes on Algebraic Topology.

I have a question regarding the following diagram from May's notes on Algebraic Topology. On pg. 7, the following diagram is given as a proof of the fact that $[f^{-1}.f]=[c_x]$. Here $f$ is a path ...
0
votes
0answers
42 views

lifting loops on surface with abelian fundamental group for decrease their self-intersection number

I found a proof of following statement (which is available here ), but I'm not sure if we must assume that fundamental group of surface $M$ is non-abelian: Lemma: Let $M$ be a compact orientable ...
0
votes
0answers
40 views

What is $H_1(A_\mathbb{C}^{top},\mathbb{Q})$

Let $A$ be an abelian variety defined over a number field. I have seen in a few papers the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ being used. I read up on the singular homology but it ...
0
votes
0answers
59 views

Link complement simply-connected if codimension $\geq 3$

In Rolfsen, page 50 says that "an easy general position argument shows that a PL link $L^k$ in $S^n$ has simply-connected complement if $n - k > 3$," where $L^k$ is a $k$-dimensional link in $S^n$. ...
0
votes
0answers
43 views

How to describe a homomorphism from a fundamental group to a finite group?

Let $S$ be a connected locally noetherian scheme with $s$ a geometric point of $S$. I read something like this: to give a surjective continuous homomorphism from $\pi_1(S, \bar{s})$ to a finite group ...
0
votes
0answers
93 views

How to compute the Lefschetz number

Given a continuous function $f: X \to X$ how do you compute: $$ \Lambda_f = \sum_{k \geq 0} (-1)^k \mathrm{Tr}(f_*|H_k(X,\mathbb{Q})) $$ which is known as the Lefschetz number. For instance let $X: ...
0
votes
0answers
41 views

Homeomorphic homogeneous 3-simplicial complexes

I have a simple question on homogeneous (i.e. made only of tetrahedron) 3-simplicial complexes. Suppose we have an homogeneous 3-simplicial complexes. Suppose we choose any couple of simplices having ...
0
votes
0answers
46 views

The relative homology of a pair $H_n(S_n, A)$.

At page 136 of Hatcher's book he says: By excision, the central term $H_n(S^n, S^n - f^{-1}(y))$ in the preceding diagram is the direct sum of the groups $H_n(U_i, U_i-x_i) \approx \mathbb{Z}$. ...
0
votes
0answers
68 views

What is “Triangulable Triad”?

I am reading George W. Whitehead's Homotopy Theory; Corollary 1.0.2 mentioned the term "Triangulable triad" without definition. May I know how it is defined?
0
votes
0answers
98 views

Finite covering space with compact spce.

Prove that if $p: \ Y \rightarrow X$ is finite covering, then if $Y$ is compact so it is X. Can someone check my attempt? :) Let $\mathcal{U}$ be any open cover of $X$. For every $x \in X$ let us ...
0
votes
0answers
98 views

Using the Hodge theorem to decompose the metric tensor

There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ...
0
votes
0answers
37 views

Understanding definition of properly discontinuous action

From Bredon, we say that the $G$-action on a space $X$ is properly discontinuous if "Each point $x \in X$ has a neighborhood $U$ such that $g(U) \cap U \neq \emptyset$" implies "$g = e$, an ...
0
votes
0answers
85 views

Classification of closed surfaces

I am doing a course in topology and is currently working on the classification theorem for closed surfaces. After realizing that every closed surface is either homeomorphic to the sphere or the sphere ...
0
votes
0answers
64 views

lifting a closed curve

Is it always true (because of covering spaces has homotopy lifting property)? loop $f$ lifts to a closed curve if and only if any curve freely homotopic to $f$ lifts to a closed curve. or we have to ...
0
votes
0answers
20 views

Why is this quotient homeomorphic to $T^2\vee S^1$?

I'm considering $X$ a closed, orientable surface of genus two with $B$ a circle in $X$ which goes around one of the handles. The picture can be found in problem 17 on page 132 of Hatcher's AT. My ...
0
votes
0answers
18 views

Suppose that complexes $K_1$ and $K_2$ have the same simplexes but different orientations. How are the chain groups $C_p(K_1)$ and $C_P(K_2)$ related?

Suppose that complexes $K_1$ and $K_2$ have the same simplexes but different orientations. How are the chain groups $C_p(K_1)$ and $C_P(K_2)$ related? I think nothing changed and they are ...
0
votes
0answers
52 views

Computing Homology Groups Systematically

I am studying for a final in an Algebraic Topology class, and I am having some trouble being able to compute homology groups for (relatively) arbitrary spaces. I understand that one possible way is ...
0
votes
0answers
59 views

Products of cells in a CW complex

Suppose I have a manifold which has a CW structure with cells $e^0 \cup e^1 \cup e^2$, where $e^i$ represents an $i$-cell. If I took the direct product of this manifold with another manifold which has ...
0
votes
0answers
16 views

How to Resolve Extension Issues in Equivariant (Co)Homology Computations

I am computing equivariant homology, which is just the usual homology of the Borel construction. I have reached a few extension issues. I have used the Gysin sequence, Leray-Serre spectral sequence, ...
0
votes
0answers
38 views

express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
0
votes
0answers
33 views

$p_n:S^1\to S^1$ is a covering

I just started learning about covering spaces and was hoping someone could fill in the details for me with this particular example. We think of $S^1\subset\mathbb{C}$ and define $p_n:S^1\to S^1$ by ...
0
votes
0answers
60 views

What is this space homeomorphic/homotopy equivalent to?

Let $X \subset \mathbb{C}^2$ be given by the equation $$|z|^2 + |w|^2=1$$ and let $A \subset X$ be given by $|z|=1,\ w=0$. This question requires me to find the relative homology groups $H_n(X,A)$, ...
0
votes
0answers
27 views

Proving $\alpha*\alpha^{-1}*p*\alpha*\alpha^{-1}$ and $p$ are homotopic

For two points $x_1$ and $x_2$ in a path-connected component, we know that their fundamental groups are isomorphic. In other words $$\pi_1(X,x_1)\cong\pi_1(X,x_2)$$ Let $\alpha$ be a path from $x_1$ ...
0
votes
0answers
30 views

Euler characteristic and free action

If $K$ is a finite simplicial complex, and $G$ acts simplicialy on $K$ with no fixed points, show $\chi(K) = |G|\cdot\chi(K^2/G)$. Could I have a hint for how to start this question? I was told to ...
0
votes
0answers
38 views

Equivalence of Cohomology groups

Suppose $n=i+j,$ with $n, i,j$ positive integers. Let $I^k$ denote the $k$-dimensional unit square. It is claimed (in Hatcher's Algebraic Topology text) that $H^i(\mathbb{R}^n, \mathbb{R}^n \setminus ...