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

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4
votes
0answers
31 views

bijection between the free homotopy classes $[S^{n},X]$ and the orbit space

I would like to prove that there is a bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_{n}(X,x_{0}) / \pi_{1}(X,x_{0})$ where the action of $\pi_{1}(X,x_{0})$ over ...
0
votes
1answer
40 views

relative homotopy groups

I study relative homotopy groups and I have a question: Let $A\subseteq X$ (not necessarily CW complex) and $\pi_{n}(X,A)$. Is it always possible to find a pointed space Y for which ...
0
votes
0answers
6 views

prove admissible monomial

Set $\mathcal{F}_k\subset P_{k-1}$, where $$\mathcal{F}_k = \{x_j^2x_{j_1}x_{j_2}\ldots x_{j_{k-3}}: 1\leqslant j_1 < j_2 < \ldots < j_{k-3}<k, \ 1 \leqslant j <k\}$$ Show that, let ...
1
vote
1answer
78 views

Cohomology Group of $CP^2 \wedge CP^2$

Calculate the cohomology group of $CP^2 \wedge CP^2$ To do this, at first I am trying to calculate the homology group and then use Universal Coefficient Theorem. To do this, at first I have ...
2
votes
0answers
14 views

Is a finite cyclic group a Poincare duality group?

I am trying to understand whether the finite cyclic group of order $n$, $C_n$ is a Poincare duality group, i.e. whether it's classifying space $K(C_n,\,1)$ is a Poincare complex. I know that the ...
5
votes
1answer
59 views

Hurewicz map factors through bordism homology

I've read in multiple sources that the hurewicz map $h \colon \pi_n(X) \to H_n(X)$ factors through oriented bordism homology. I'm particularly interested in the injectivity of the map $h \colon ...
3
votes
1answer
309 views

Tangent bundle of Grassmann manifold

I have to prove that the tangent bundle of Grassmann manifold $G_n(\mathbb{R}^{n+h})$ is isomorphic to $\operatorname{Hom}(\gamma^n(\mathbb{R}^{n+k}),\gamma^\perp)$, with $\gamma^{\perp}$ is the ...
2
votes
1answer
31 views

Stiefel-Whitney Classes of a submanifold

Suppose we have a manifold $M$ that is the product of $k$ copies of real projective space, say $$ M = \prod_{i=1}^k \mathbb{R} P^{n_i}.$$ Then $H^*(M; \mathbb{Z}/2) = \mathbb{Z}/2 [ \alpha_1, \dots, ...
2
votes
1answer
61 views

Section 22 in Munkres' TOPOLOGY, 2nd edition: How to establish this equivalence?

Let $X$ and $Y$ be topological spaces; let $p \colon X \to Y$ be a surjective map. Then $p$ is said to be a quotient map provided a subset $U$ of $Y$ is open in $Y$ if and only if $p^{-1}(U)$ is ...
3
votes
2answers
43 views

Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way?

Let $X$ be a topological space space and $X=\cup_{i\in I} U_i$ a covering of $X$ by open subsets $U_i\subseteq X$. Is it true that $$ \operatorname{colim}\left(\coprod_{(i,j)\in I\times I} ...
2
votes
2answers
38 views

Prove that exist bijection between inverse image of covering space

Let $B$ be path-connected and $p:E\to B$ covering map (with $E$ as covering space). Prove that $\forall a,b\in B$ exist 1-1 injection correspondence between $p^{-1}(a)$ and $p^{-1}(b)$ I thought ...
2
votes
1answer
53 views

Working with homomorphisms and de Rham cohomology.

Here’s my question: Let M be a connected, compact, orientable, smooth n-manifold ($ n \in \mathbb{N}_{\geq 2} $). Let V be a neighborhood of p diffeomorphic to $\mathbb{R}^n$ and let U = M \ {p}. ...
2
votes
0answers
56 views

$\mathsf{Top}$ with proper maps has products.

In I.M. James' General Topology and Homotopy Theory, he presents proper maps before introducing compact sets, by defining $\phi:X \to Y$ to be proper iff $\phi \times \text{Id}_T$ is closed for all $T ...
2
votes
1answer
32 views

Join of closed embeddings is a closed embedding

An exercise from James's book General Topology and Homotopy Theory asks the reader to prove that if $\phi_1:X_1 \to Y_1$ and $\phi_2:X_2 \to Y_2$ are closed topological embeddings, then $\phi_1 * ...
5
votes
2answers
73 views

Finding the de Rham cohomology of an open subset of $ \Bbb{R}^{n} $ minus a point.

Here’s my question: Let $ n \in \mathbb{N}_{\geq 2} $. Suppose that $ U \subseteq \Bbb{R}^{n} $ is an open set and that $ x \in U $. Then show that $$ {H_{\text{dR}}^{n - 1}}(U \setminus \{ x ...
4
votes
1answer
26 views

Completing compact surfaces with boundary to closed surfaces in $\mathbb R^3$

My question is whether any compact smooth surface in $\mathbb R^3$ (with smooth boundary) can be completed to a closed smooth surface in $\mathbb R^3$ without boundary? It is easy to complete it to an ...
3
votes
1answer
19 views

how to compute the de Rham cohomology with compact support of a mobius strip

I am having problem computing the de Rham cohomology with compact support of an open mobius strip,it's aquestion from Bott's book, and Bott said its cohomology is identically zero which can be ...
-4
votes
0answers
57 views

Cup product Structure of $X \vee Y $

Suppose $\alpha \in H^*(X)$ and $\beta \in H^*(Y)$ are of positive degrees. Show that $\alpha\beta=0$ in $H^*(X \vee Y)$. I am unable to show that. I think $\alpha\beta=0 $ because intersection of ...
0
votes
1answer
39 views

Show that $p:SO_3 \to\mathbb S^2 $ defined as $p(A)=Ae_1$ is a fibre bundle

Show that $p:SO_3 \to\mathbb S^2 $ defined as $p(A)=Ae_1$ is a fibre bundle. I know that $SO_3$ acts on $\mathbb S^2$ transitively saying that $p$ is onto.I have a problem with local ...
1
vote
1answer
41 views

What exactly is the Kahler class of a torus?

Is there an intuitional way to understand what the Kahler class of $T^2$ actually is? It would be extremely useful to me if you could provide me some intuition behind it!
0
votes
0answers
50 views

Cohomology ring of $S^3 \setminus A $ and $S^3 \setminus B $,where $A$ is union of two once linked circle and $B$ is union of two unlinked circles

Suppose $A$ is union of two once linked circles in $S^3 $ and $B $ is union of two unlinked circles.show that $S^3 \setminus A $ and $S^3 \setminus B$ have same cohomology group but not same ...
1
vote
1answer
44 views

computing Lefschetz number

We have a fixed point theorem which says that : Let $X$ be a compact polyhedron, $f:X\rightarrow X$ be a continuous map. If $L(f)\neq 0$ then $f$ must have a fixed point. (Lefschetz number is ...
-2
votes
0answers
15 views

lemma( lifting- cantor path )? [on hold]

Definition: Let $p:E\rightarrow B$ be a map. If $f:X\rightarrow B$is a map, a lifting of is a map $\widetilde{f}:X\rightarrow E$ such that $p\circ \widetilde{f}=f$ ¿TRUE or FALSE? "Let $C$= Cantor ...
3
votes
2answers
81 views

Finding the Fundamental Groups of Some Modular Spaces

I'm looking to compute the fundamental group of a couple of different quotients of the $n$-torus. The first of these I'm interested is the space $\mathbb{T}^n/S_n$ where the symmetric group $S_n$ ...
4
votes
1answer
71 views

If $f:S^1\to S^1$ doesn't have any fixed point then it is homotopic to the identity

How to show that every continuous function $f:S^1\to S^1$ without fixed points is homotopic to the identity? (without using homology nor the concept of degree).
0
votes
0answers
27 views

thorough proof of Mayer-Vietoris implies Excision

Where could I find a very complete proof of how the Mayer-Vietoris sequence implies the Excision theorem? I've read a few proofs, but they always leave out the details! Thank you!
2
votes
3answers
45 views

simply connected covering of a path connected space (II)

Let $p:\overline{X}\rightarrow X$ be a simply connected covering of a path connected space $X$ and $A\subset X$ be a path connected set. Show that the inclusion induced homomorphism $i_{\sharp} : ...
4
votes
1answer
63 views

Milnor's definition of bundle map in “Characteristic Classes”

In chapter 3 of "Characteristic Classes", Milnor defines bundle maps, requiring them to map fibers isomorphically onto fibers. Why not merely require homorphisms on fibers? (e.g., for the given ...
-1
votes
0answers
28 views

Curve concatenation in manifolds.

I am having difficulty understanding what is going on geometrically when you add together multiples of curves (1-chains) in a differentiable manifold. Say we have two curves $A$ and $B$ together with ...
10
votes
1answer
139 views

Vector space identity from Chow's “You Could Have Invented Spectral Sequences”

In Chow's You Could Have Invented Spectral Sequences (3rd page, left column) appears the following isomorphism of vector spaces: $$\frac{Z_d}{B_d}\cong \frac{Z_d+C_{d,1}}{B_d+C_{d,1}}\oplus ...
8
votes
1answer
4k views

Calculating fundamental group of the Klein bottle

I want to calculate the Klein bottle. So I did it by Van Kampen Theorem. However, when I'm stuck at this bit. So I remove a point from the Klein bottle to get $\mathbb{Z}\langle a,b\rangle$ where ...
5
votes
2answers
312 views

$\pi_1$ and $H_1$ of Symmetric Product of surfaces

Let $X=Sym^d(\Sigma_g)$ be the d-fold symmetric product of a genus-g surface, $d\ge 2$. Is there / what is a (quick simple) way to see that $\pi_1(X)$ is abelian? The link in the comments ...
6
votes
1answer
90 views

Homotopic equivalence for $S^5$ without three $S^1$

Consider a standard embedding of $S^5$ in $\mathbb R^6$: $S^5: \; x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 = 1.$ And consider three circles, which are sections of $S^5$ by $x_1 x_2$, $x_3 x_4$, ...
1
vote
1answer
119 views

If a Klein bottle covers a manifold $M$, then $M$ is the Klein bottle

I have to prove that if a Klein bottle covers a manifold $M$, then $M$ is the Klein bottle. Any suggestions? Thanks.
2
votes
2answers
125 views

$K$-theory exact sequence.

Let $Y$ be a closed subspace of a compact space $X$. Let $i:Y \to X$ the inclusion and $r:X \to Y$ a retraction ($r \circ i = Id_Y$). I have to prove that exists this short exact sequence $$ 0 \to ...
0
votes
0answers
40 views

Fundamental group of the mapping cone of a loop

You can read in Wikipedia the following: Given a space X and a loop $\alpha\colon S^1 \to X$ representing an element of the fundamental group of $X$, we can form the mapping cone $C_α$. The effect of ...
0
votes
0answers
30 views

Betti numbers over unital rings [on hold]

Is the following statement correct? Given a manifold $M$. If $H_1(M,\mathbb Z)$ is a finite cyclic group, then the first $R$-Betti number $b_1(M,R)$ is bounded from above by $1$ for every unital ring ...
15
votes
1answer
369 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
3
votes
1answer
35 views

Calculating the homology groups of a simplicial complex using a Mayer-Vietoris sequence

I'm trying to calculate the homology groups for a simplicial complex $X$, which is a union of subcomplexes $X_1$ and $X_2$ which are both combinatorially equivalent to cones. This is the information I ...
2
votes
1answer
85 views

Examples with zero first Stiefel-Whitney class and nonzero second Stiefel-Whitney class

What's the simplest/most concrete vector bundle you can think of that has zero first Stiefel-Whitney class but non-zero second? That would be the simplest space that doesn't have spinors. (See Spin ...
1
vote
1answer
39 views

Write down a map $f$ from the torus $T$ to itself such that the induced map $g:H_1(T) \to H_1(T)$ is given by the matrix ( 1 1 : 0 1)

I think $f(x,y)=(x,x+y)$. suppose $f(x,y)=(x,x+y)$.then I am looking at the action of $g$ on the generators of $H_1(T)$. but I can't show that.
1
vote
1answer
37 views

Injection of the mapping cone of $z^2$

We define the mapping cone of $f:S^1\to S^1=:Y$, $f (z)=z^2$ as the quotient space of $S^1\times [0,1]\sqcup Y$ where $(z,0)$ and $(z',0)$ are identified and where $(z,1)$ and $f(z)$ are identified ...
11
votes
2answers
295 views

Does the ham sandwich theorem hold for dividing objects into thirds?

The ham sandwich theorem states that given $n$ measurable "objects" in $n$-dimensional space, it is possible to divide all of them in half (with respect to their measure) with a single ...
3
votes
1answer
60 views

Understanding Hatcher's proof for $\chi(M)=0$ for non-orientable manifolds $M$ of odd dimension

In the Corollary 3.37 Hatcher proves that for a closed odd-dimensional manifold $M$, its Euler characteristic is zero. The first part of the proof deals with orientable manifolds, and uses Poincare ...
-1
votes
2answers
33 views

Covering maps question [closed]

I need to show that, given $p_1:Y_1\rightarrow X $ and $p_2:Y_2\rightarrow X $ two covering maps locally pathwise connected, and there exists $\alpha$ a covering homomorphism between p1 and p2, then ...
1
vote
1answer
200 views

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 ...
3
votes
2answers
70 views

Is composition of covering maps covering map?

In Munkres book, composition of covering maps is covering map when $r^{-1}(z)$ is finite for each $z$ in $Z$ where $q : X\to Y$ , $r:Y\to Z$ are the covering maps. I tried hard to find an example that ...
4
votes
1answer
134 views

4-manifold: $0$-handle $\cup$ $2$-handles along a framed link in $S^3$ (intersection form = linking matrix = presentation matrix of $H_1(\partial M)$)

Let $L$ be a framed link in $S^3$, consisting of framed knots $L_1,\ldots,L_m$. Let $A=[a_{ij}]\in\mathbb{Z}^{m\times m}$ be its linking matrix, with $a_{ii}=$ framing of $L_i$ and $a_{ij}=$ linking ...
1
vote
0answers
83 views

Principal $G$-bundles as pull back bundles. [duplicate]

Let $G$ be a compact Lie group and consider a $G$-universal bundle $\pi: EG \to BG $ where $BG$ is the classifying space for the group $G$ and the bundle $\pi: EG \to BG $ is defined as the principal ...
2
votes
1answer
57 views

Build sheaf from stalks

If I have a topological space $T$ and for each $p \in T$ I have an object $A_p$ in some category $\mathscr{A}$, then how can I define a sheaf out of this? In other words can I build a sheaf with ...