# 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.

20,682 questions
Filter by
Sorted by
Tagged with
87 views

### Cohomology ring of symmetric products (of manifolds)

Let $S_g$ be a closed, orientable surface of genus $g$ (new notation in light of the first comment). I am looking for results to determine explicitly the (co)homology groups and/or cohomology ring ...
1 vote
22 views

### How to compute the monomial matrices?

In GTM227 Combinatorial Commutative Algebra, Miller defined the monomial matrix to represent a map between two $\mathbb{N}^n$-graded free $S$ modules. Its columns are labeled by sourse degrees $a_p$ ...
30 views

### Examples of CW-complexes wich are 1-acyclic but no simply conected.

Hurewicz theorem states that if $X$ is a simply-connected CW-complex then $X$ is $(n-1)$-connected if and only if it is $(n-1)$-acyclic and that in this case $\pi_n(X)=H_n(X)$. Moreover, it is also ...
93 views

31 views

### The complementaries of homologous objects in $S^n$ are homologous?

Suppose $A$ and $B$ are two subsets of the $n$-sphere $S^n$ that have isomorphic finitely-generated homology groups. Do $S^n\setminus A$ and $S^n\setminus B$ also have isomorphic homology groups? I ...
1 vote
95 views

### Contractibility of the based path space.

Let $X$ be a topological space and $x_0 \in X.$ Consider the based path space $$\mathcal P_{x_0} (X) : = \left \{\gamma : [0,1] \xrightarrow{\text {continuous}} X\ \bigg |\ \gamma (0) = x_0 \right \}$$...
14 views

### Spectral sequence for a truncated semi cosimplicial space

Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a ...
24 views

### How to draw Caley complex for any group

I am reading Algebraic Topology by Allen Hatcher. I come to know for any group $G$ ,we can make an universal cover $X$ of $X/G$ by properly discontinuous action . There is a paragraph mentioned in ...
1 vote
23 views

### Relationship between compactly supported homology and homology with orientation sheaf

Let $M$ be noncompact and non-orientable manifold of dimension d. Is it true that $H_{c}^{d-i}(M;Q)$ and $H^{i}(M;Q^{w})$ are isomorphic, where $Q^{w}$ is orientation sheaf and $Q$ is a field of ...
63 views

### Topology of Mobius Strip

The mobius strip $M$ is topologically distinct from the cylinder $S^1\times I$ where $I$ is a finite segment of $\mathbb{R}$ (namely, one cannot be deformed into the other without cutting and pasting)....
15 views

### If $H$ is a closed subgroup of a Lie Group $G$, and $p:P\to B$ a principal G-bundle. How to show that $q:P\to P/H$ is a principal $H$-bundle?

Let $G$ be a Lie group and $H$ a closed subgroup of $G$. Suppose we are given a principal $G$-bundle $p:P\to B$, how to show that the quotient $q:P\to P/H$ is a principal $H$-bundle? Where $P/H$ ...
47 views

### Hatcher 1.1.18 - sufficient solution?

Could someone check the following solution to Hatcher 1.1.18? I'm wondering whether the arguments in the end using van Kampen are sufficient in normal proof-writing in algebraic topology. Problem: ...
80 views

### Show that the map $s : X \longrightarrow \widetilde X$ is well-defined.

Let $p\ \colon \widetilde X \longrightarrow X$ be a covering map such that any loop $\gamma$ in $X$ based at $x_0 \in X$ lifts uniquely to a loop based at $\widetilde {x_0} \in p^{-1} (\{x_0\}).$ ...
1 vote
63 views

### Does path connectedness simplify the proof of the LES in reduced homology?

I'm reading Weintraub's Fundamentals of Algebraic Topology, in which there is an exercise (3.4.6 in the first edition) that wants us to show that for every path connected subspace $A$ of a path ...
1 vote
63 views

### What was the intended, more elementary solution to Hatcher $2B.5$? On isomorphisms in homology between a sphere complement and a subsphere

I solved Hatcher's exercise $2B.5$ but I wonder if there is a more elementary approach. Paraphrasing, and removing trivial or vacuous cases, the exercise is this: Suppose $n\ge1$ and $0\le k\le n-1$ ...
1 vote
33 views

### Bilinear pairing on homotopy groups

Let $X,Y,Z$ be pointed spaces, and $f:X \wedge Y \rightarrow Z$ a map. Then, $f$ induces a bilinear pairing $\pi_n(X) \times \pi_m(Y) \rightarrow \pi_{n+m}(Z)$ ($n,m \geq 1$). I see what the pairing ...
1 vote
51 views

### Mapping cylinder being mobius band - example 1.35 in Hatcher

A lot of the geometric examples in Hatcher with abstract identification spaces I have a hard time visualizing. In the following example, Hatcher says that the mapping cylinder of $z \to z^2$ is the ...
1 vote
110 views

Let $G$ be a finite group and $X$ a pointed $G$-space. The assignment $G/H\to \pi_n(X^H)$ should define a Mackey functor. I am trying to figure out what the transfers and restrictions are. If $H\... 1 vote 1 answer 103 views ### Is the quotient space of a homogeneous space induced by a free group action s.t. the quotient map has a right inverse homogeneous? Given is a topological space$X$and a group$G \leq$Aut($X$) with the property: for$\lambda \in G$and$x \in X$,$\lambda(x)=x \Rightarrow \lambda=id_X$, also satisfying that the quotient map$q:X ...
40 views

I am struggling to understand part of the top answer here: Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell A space $A$ is glued onto a general topological space $X$ ...
63 views

### A cell complex homeomorphic to $S^n$

Proposition. If $X$ is a finite $n$-dimensional cell complex, such that each $(n-1)$-cell is contained in exactly two $n$-cells, then $X$ is homeomorphic to $S^n$. This statement sounds quite ...
165 views

27 views

### What is the mean of "the isometries that identify the sides of polygon"?

Picture below is from the 167th page of do Carmo's Riemannian Geometry. I don't know the mean of "the isometries that identify the sides of $P$". My English is poor. I know the process of ...
52 views

### Nerve Theorems for Open Coverings [closed]

There are numerous Nerve theorems that come down to something like this: For an open cover U of a space X, let N(U) be the nerve. Then under certain conditions, N(U) is homotopy equivalent to X. The ...
36 views

47 views

### Representation of topological K-theory via Brown representability

We know that topological K-theory is a generalized cohomology theory, and reduced K-theory is a reduced cohomology theory. Thus, both are representable with a sequence of pointed homotopy functors, ...
33 views

### Limitations of Ω-Spectra

Right now I am interested in the "stabilization" endofunctor of the category of ∞-groupoids sending an object $X$ to $\text{colim } Ωⁿ Σⁿ X$. This colimit is related to $∃Y:X≅ΩY$. In Ω-...
74 views

### Explicitly understanding the map representing first cohomology

Let $M$ be a closed smooth manifold. It is well-known that there is a bijection between the cohomology group $H^n(M;G)$ and $[M,K(G,n)]$, where $G$ is a group and $K(G,n)$ is an Eilenberg-Maclane ...
81 views

### Spaces with two non-trivial homotopy groups

I'm wondering if there is any elementary example of a space with precisely two non-trivial homotopy groups. Let $X$ be a connected CW complex with precisely two non-trivial homotopy groups $\pi_p$ and ...
98 views

### Hatcher 1.2.4 solution?

Problem 1.2.4: Let $X \subset \mathbb{R}^3$ be the union of $n$ lines through the origin. Compute $\pi_1 (\mathbb{R}^3 - X)$. I have the following solution. Would someone be so kind as to check ...
61 views

### A Problem With Simplicial DeRham Cohomology

I am getting a contradiction by calculating the simplicial DeRham complex of an arbitrary manifold and getting it to be trivial. I also get a similar contradiction using the Godement resolution for ...
Suppose that $K \subset G$ is a topological subgroup and this inclusion is a homotopy equivalence (so somewhat stronger than what's written in the title). I'm not assuming compactness but am happy to ...
Let us consider a set of $N$ vectors, $\mathbf{h}_1, \mathbf{h}_2, \cdots, \mathbf{h}_N$, such that $\mathbf{h}_i \in \mathbb{R}^{M}$, $\forall i$, with $M < N$. Let us also consider the space \$\...