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

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2
votes
0answers
16 views

Classifying Covering Spaces using First Cohomology

I am familiar with the classification of covering spaces of a space $X$ in terms of subgroups of $\pi_1(X)$ (up to conjugation). However, if $X$ is a manifold, I know that $H^1(X; G)$ classifies ...
3
votes
1answer
60 views

Property of second Steifel-Whitney class?

Let $M$ be manifold, $n = 4$. Is $w_2$ special in in the regard it's the only thing of $H^2(M, \mathbb{Z}_2)$ where $w_2 \cup \tau = \tau \cup \tau$, $\tau \in H^2(M, \mathbb{Z}_2)$ or not? I wondered ...
2
votes
1answer
21 views

Local isometry between simply connected manifolds

Suppose $D:\tilde{M}\rightarrow N$ is a local diffeomorphism between two simply connected smooth manifolds $\tilde{M}$ and $\tilde{N}$. $D$ is onto. In the case of $D$ being a covering map, it follows ...
2
votes
2answers
23 views

Are there any nontrivial second Hurewicz homomorphisms for familiar compact 6-dimensional manifolds?

Based on several computations I have done, it seems that the second Hurewicz homomorphism $$h:\pi_{2}(X)\rightarrow H_{2}(X;\mathbf{Z})$$ has a habit of being trivial. For instance, this seems to be ...
0
votes
0answers
49 views
+50

Topology of Isolated Singularities

In Singular Points of Complex Hypersurfaces, Milnor shows that if $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$, holomorphic, has an isolated singularity at the origin, then $f^{-1}(\epsilon) \cap ...
9
votes
3answers
692 views

Eckmann-Hilton and higher homotopy groups

How does the Eckmann-Hilton argument show that higher homotopy groups are commutative? I can easily follow the proof on Wikipedia, but I have no good mental picture of the higher homotopy ...
63
votes
10answers
3k views

Explain “homotopy” to me [on hold]

I have been struggling with general topology and now, algebraic topology is simply murder. Some people seem to get on alright, but I am not one of them unfortunately. Please, an answer I need is ...
6
votes
0answers
61 views
+100

Constructing $\mathbb{P^n}$ “bundle” with different $n$

Can we find a complex/smooth manifold and map $f\colon X\to\mathbb{C}^3$ such that it is a $\mathbb{P}_\mathbb{C}^m$ bundle over linear subspace $\mathbb{C}^1$ and $\mathbb{P}_\mathbb{C}^n$ bundle ...
0
votes
1answer
40 views

Covering map in the context of Riemann Surfaces and Algebraic Topology

I am taking a course in Riemann surfaces and our lecturer has warned us that the definition of covering maps in the context of Riemann surfaces is strictly weaker than the ones used in Algebraic ...
3
votes
2answers
175 views

What topological space do I obtain by gluing these edges?

I'm gluing the edges of a square together with the caveat that there's a "fold" down the middle. I think this produces sort of a sphere with four "pinches". I'm wondering if my intuition is correct ...
1
vote
0answers
33 views

Hatcher deduce ring structure on $H^\ast(\mathbb{R}P^\infty;\mathbb{Z})$

So in Hatcher they deduce the ring structure of $H^\ast (\mathbb{R}P^\infty;\mathbb{Z})$ by looking at the map $\mathbb{Z}\rightarrow\mathbb{Z}_2$, which induces maps on ...
2
votes
0answers
62 views

Topology of the complex curve $x^4+y^4=1$

How do you realize that the complex curve $x^4+y^4=1$ looks topologically like three tori glued together with four points at infinity?
1
vote
0answers
22 views

triangulating torus using simplices

i am currently pursuing a course in basic homology theory.i am really stuck in how to triangulate a torus by using simplicial complexes . in every book, a diagram is given but it does not define that ...
1
vote
1answer
53 views

Is there a nontrivial fiber or principal bundle over $S^3$?

Is there a nontrivial fiber or principal bundle over $S^3$?I know that, by a paper of Steenrod,see the link below, every sphere bundle on 3- sphere is trivial but what about arbitrary fiber ...
1
vote
1answer
22 views

Finding triangulations of spaces.

I am currently pursuing a course in basic homology theory and i am finding it really difficult to find the triangulation of spaces. I know that a triangulation of a topological space $X$ is a ...
7
votes
0answers
86 views

Is every topological group the topological fundamental group of an space?

The fundamental group $\pi_{1}(X)$ of a path connected topological space $X$ is the image of $Hom(S^{1},X)$. So the fundamental group can be topologized with quotient topology where ...
1
vote
1answer
36 views

A lemma from Hilton & Stammbach's book A Course in Homological Algebra

In orde to prove the set of equivalence classes of extensions of $A$ by $B$ is a contravariant functor of the first component and covariant functor of the second. The authors give us three lemma. I'm ...
2
votes
2answers
23 views

The configuration space of directed great circles on the sphere

I am reading a proof of Pu's inequality in Katz' book "Systolic geometry and topology". In section 6.4, he describes a fibration $q : SO(3) \to S^2$ which I can't quite figure out. First, he thinks ...
8
votes
6answers
100 views

Examples of global properties that don't arise from local knowledge

Let $M$ be a smooth manifold. As an example of a global property that arises from local data we know that if $(M,g)$ is a compact surface without boundary then the Euler characteristic is given by $$ ...
1
vote
2answers
44 views

Equivariant cohomology via equivariant sheaves

Ordinary cohomology of topological space $X$ are known to be the cohomology of constant sheaf. Question Is there analogous description for equivariant cohomology? More precisely. Consider category ...
1
vote
1answer
23 views

Do path homotopy classes of concatenated paths have a middle fixed point?

If $[a]$ and $[b]$ are path homotopy classes, then $[a]\cdot[b]$ is defined as $[a*b]$, where $a*b$ is defined as the concatenation of the paths $a$ and $b$. Let us say $a(1)=b(0)=p$. Then does each ...
1
vote
0answers
21 views

Quotient topology from Delta complex,

A $\Delta$-complex structure on a space X is a collection of maps $\sigma_{\alpha} : \Delta^n \rightarrow X$ with n depending on index $\alpha$ such that 1)The restriction ...
4
votes
1answer
38 views

Does $X^{C_2} \simeq * \simeq X/{C_2}$ imply $X \simeq *$?

What the title says. Let $C_2$ be the cyclic group of order 2, and $X$ be a topological space with a $C_2$-action (acting continuously) such that both the quotient space $X/{C_2}$ and the subspace of ...
2
votes
2answers
62 views

Long exact sequence of a triple: working through the geometry

Suppose $X$ is a topological space with subspaces $X \supset U \supset A$ such that $U$ deformation retracts onto $A$. We know that $H^*(X,U) \cong H^*(X,A)$--one way to see this is to take the long ...
3
votes
1answer
37 views

There are no nonzero cocycles on $U$ vanishing on a def. retract of $U$

Playing around with cochains, I think I showed the following: Proposition: Suppose $U$ deformation retracts onto its subspace $A$, and suppose $\varphi \in C^k(U)$ is a singular cocycle which ...
2
votes
2answers
83 views

Problem 10 of Section 1.2 from Hatcher.

Problem. Consider two arcs $\alpha$ and $\beta$ embedded in $D^2\times I$ as shown in the figure. The loop $\gamma$ is obviously nullhomotopic in $D^2\times I$, but show that there is no ...
3
votes
3answers
456 views

$M \times N$ orientable if and only if $M, N$ orientable

For two manifolds $M$ and $N$ I'm trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then ...
1
vote
1answer
318 views

How to show $S^n$ is not contractible without using Homology..

I know the prove $S^n$ is not contractible using homology.But I don't know how to prove it from definition of contractibility.Can anyone help me in this direction? Thanks.
0
votes
0answers
32 views

Why is the unit disc not a topological surface? [duplicate]

I am trying to prove that the unit disc $D^2$ is not a topological manifold. Clearly it is Hausdorff and second countable, so I think I should show that it is not locally Euclidean. The following is ...
4
votes
2answers
577 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 ...
1
vote
1answer
78 views

Injectivity in the zero homology

I'm struggling with following step in an excercises about Mayer-Vietoris sequences: In one step the solution says this map is injective since $A \cap B$ is path-connected: $$ H_0(A \cap B) ...
1
vote
2answers
56 views

Long exact sequence of $(I \times Y, \partial I \times Y)$.

There's a section in chapter 3 in Hatcher's Algebraic Topology where he talks about the long exact sequence of the pair $(Y \times I, Y \times \partial I)$, where $Y$ is any topological space. The ...
14
votes
5answers
2k views

Poincare Duality Reference

In Hatcher's "Algebraic Topology" in the Poincare Duality section he introduces the subject by doing orientable surfaces. He shows that there is a dual cell structure to each cell structure and it's ...
1
vote
3answers
66 views

How do we view natural transformations as functions

1.The definition asserts that natural transformation is a map of two functors. However, from the definition, given tow functors $F,G:C,D$, we associate every element $x$ in $Obj(C)$ a morphism $F(x) ...
2
votes
0answers
46 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
2
votes
1answer
34 views

Spaces homotopy equivalent to finite CW complexes

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of ...
1
vote
1answer
21 views

Is the transfer homomorphism in cohomology surjective?

Let $p:\widetilde{X} \to X$ be an $n$-sheeted covering space. Consider a singular simplex $c:\triangle^k \to X$, because the simplex is simply-connected, there exist $n$ different lifts ...
0
votes
1answer
22 views

Homotopy $I^2 \rightarrow S^1$ lifting lemma proof

In case of a homotopy $h: I^2 \rightarrow S^1$ we can define lifting as such an $\tilde{h}: I^2 \rightarrow \mathbb{R}$ that $e^{i\tilde{h}}=h$. The existence of $\tilde{h}$ requires a proof. A way to ...
0
votes
1answer
11 views

Regarding embeddings and homological/cohomological injectivity

Possibly low-level question here: if one has an embedding $M\hookrightarrow X$, call it $\iota$, then this is of course an immersion, so the differential maps $\iota_{\ast}:T_{p}M\rightarrow ...
0
votes
1answer
38 views

Prove closed disc $D^n$ is homeomorphic to the cone $CS^{n-1}$

I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$. For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ ...
0
votes
1answer
33 views

Is torus w. disc removed homotopic to klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know f and g are homotopic if they represent: ...
1
vote
1answer
20 views

strange implication of addivity axiom in homology theory

Let $H$ be a homology theory satisfying Eilenberg-Steenrod axioms and $X$ an arbitrary topological space. We can write $X$ as a disjoint union of its points $$X= \coprod_{x \in X}{\{x\}}$$ Now the ...
0
votes
0answers
35 views

Using transfinite induction to compute fundamental groups

I want to compute the fundamental group of a space containing comb space along with boundary of $I \times I$ where I is $[0,1]$. Here is my attempt using transfinite induction and Van Kampen. Let's ...
4
votes
1answer
106 views

Fundamental group and path-connected

Let $p:E \rightarrow B$ be a covering space, $E$ and $B$ are path-connected. Let $A$ be a path-connected subset of $B$ . How to use fundamental group to give a sufficient and necessary condition to ...
1
vote
0answers
38 views

Path space of suspension

Let $X$ be a pointed homotopy type (of a CW complex) and let $G = \Omega \Sigma(X)$ be the loop space of the suspension. Let $P$ denote the homotopy pushout of the diagram $$ G \gets G \times X \to ...
1
vote
1answer
34 views

Fundamental Property of Regular CW Complexes

$\newcommand{\R}{\mathbf R}$ For a cell $e$ in a CW complex, we write $\partial e$ to denote $\bar e-e$. Note that $\partial e$ may not be the topological boundary of $e$ in $X$. A CW complex ...
0
votes
2answers
70 views

Non-abelian fundamental group on a path-connected space

I am doing a self-study of algebraic topology, and am having some difficulties comprehending the idea of a non-abelian fundamental group on a path connected space. (See for example Hatcher Exercise ...
4
votes
1answer
410 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
3
votes
0answers
28 views

multiplication in H-space and loopspace of the H-space

Let $X$ be an $H-space$, and let a multiplication $,\cdot,$ be given, associative up to homotopy. Let $\Omega X$ be the loopspace of $X$ based at the identity and let the multiplication $ \circ $ on ...
0
votes
0answers
30 views

A problem of the definition of relative homology

This is a lemma on the textbook algebraic topology a first course (Greenberg). I supposed that they are equal, not isomorphism. Proof:$$H_q(X,A)\cong Z_q(X,A)/B_q(X,A)$$ I am wondering $Z_q(X,A)$ is ...