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

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2
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0answers
27 views

Do complete Hopf algebras have an antipode?

I am reading Quillen's paper on rational homotopy theory. In appendix A of this paper Quillen defines a notion he calls complete Hopf algebras. These are certain cocommutative bialgebra structures ...
3
votes
2answers
45 views

Example of induced homomorphism in algebraic topology

I would like to understand what induced homomorphism are, as they appear in the definition of the Mayer-Vietoris sequence. Since an homology group $\tilde{H}_n$ is a quotient group defined as ...
0
votes
1answer
32 views

With field coefficients homology and cohomology coincide

With field coefficients the universal coefficients theorem takes the form: $$H^n(X;F)=Hom_{F-modules}(H_n(X;F),F)$$. Now in all computations I have seen with field coefficients we have ...
0
votes
1answer
24 views

Homology and Reduced homology coincide on non trivial pair.

In Hatcher page 118, he says that There is a completely analogous long exact sequence of reduced homology groups for a pair $(X;A)$ with $A\not = \emptyset$ ; This comes from applying ...
0
votes
0answers
39 views

the functor Ext repairs the inexactitude of Hom on the right

For a short exact sequence of R -modules: $ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0$. Prove that: the following sequence is exact $0 \rightarrow \mathrm{Hom} ...
2
votes
1answer
58 views

Computing Klein bottle's cohomology ring in $\mathbb{Z}$

Well I've been struggling with this one. This is the picture of the Klein Bottle. It has two triangles (U upper, V lower), three edges (the middle one is "c") and only one vertex repeated 4x. So my ...
2
votes
1answer
69 views

Would this be a homology theory?

Consider a manifold $M$, and denote by $\Delta _p M$ the set of all submanifolds of dimension $p$ (with or without boundary) of $M$. Define $G_pM$ to be the free abelian group generated by $\Delta_p ...
8
votes
1answer
101 views

different definitions of Hopf algebras

(i). In the book Algebraic Topology, A. Hatcher, p. 283, the notion Hopf algebra is defined as follows: (ii). However, in the book Bialgebras and Hopf algebras, J.P. May, the notion Hopf algebra is ...
2
votes
1answer
16 views

Simplicial homology: chain group with basis open n-simplices vs. chain group with basis closed n-simplices

In his Algebraic Topology book, Hatcher defines the chain groups for simplicial homology as free abelian groups with basis the open $n$-simplices of some simplicial complex X. Is there any ...
1
vote
1answer
9 views

Name for map associated with simplicial complex

Given a simplicial complex $\Delta$, implied by the construction process there are associated maps sending euclidean standard simplices into the simplicial complex $\imath: \Delta^n \to \Delta$. What ...
5
votes
3answers
120 views

On defining homology groups

I have been trying to understand what homology groups are "talking about," and now I am wondering if the following works as a definition of homology. But first, some illustration of what it is ...
4
votes
1answer
59 views

$w(\mathbb{R}P^q) = 1$ if and only if $q = 2^k - 1$ for some $k$

How do I show that $w(\mathbb{R}P^q) = 1$ if and only if $q = 2^k - 1$ for some $k$?
5
votes
1answer
85 views

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$. I need to show that a homotopy equivalence between them doesn't exist, but it seems like the homology groups of the spaces ...
5
votes
1answer
40 views

Why “singular” in “singular homology/cohomology”?

As the title suggests, I'm curious to know whether there is any reason why the word "singular" appears in "singular homology/cohomology".
4
votes
1answer
47 views

isotopy equivalence between manifolds

The definition below is from Encyclopaedia of Mathematics: Volume 6. Question: For any $n\geq 1$, is the $n$-dimensional closed cube $$[0,1]^n=[0,1]\times [0,1]\times\cdots \times[0,1]$$ isotopy ...
5
votes
0answers
76 views

Minimum number of sets required for a good open cover

A good open cover of a topological space is an open cover such that all open sets in the cover, and all finite intersections of open sets in the cover are contractible. For example, $S^2$ has an ...
3
votes
0answers
31 views

Homology of connected sum of CW-complexes

Let $X$ and $Y$ be finite and connected CW-complexes of dimension $n$ with exactly one $n$-cell. Then we can define their connected sum $X\#Y$ just like in the manifold case: extract an ...
1
vote
1answer
71 views

Is simply connectedness is a topological property?

A topological space $X$ is called simply-connected if it is path-connected and any continuous map $f:S^{1} \to X$ (where $S^1$ denotes the unit circle in Euclidean 2-space) can be contracted to a ...
1
vote
1answer
39 views

Existence of Moore spaces for modules over commutative rings.

Let $R$ be a commutative ring, $A$ a $R$-module and $n$ a natural number. Does there exist a CW complex $M(A,n)$ with $\tilde{H}_i(M(A,n),R)=0$ if $i\neq n$ and $\tilde{H}_n(M(A,n),R)\cong A$ as ...
4
votes
0answers
54 views

Composition, fibrations?

Let $p: D \to B$ and $q: E \to B$ be fibrations and let $f: D \to E$ be a map such that $q \circ f = p$. Suppose that $f$ is a homotopy equivalence. My question is, does it follow that $f$ is a fiber ...
2
votes
2answers
40 views

Dimension of the $0^{\text{th}}$ de Rham cohomology group of $U$

Let $U\subset\mathbb{R}^n$ be an open set. I proved that $$ H^0_{DR}(U):=\frac{\text{closed forms}}{\text{exact forms}}=\{f\in C^{\infty}(U):\,f\,\,\text{is locally constant} \} $$ I have to show ...
1
vote
1answer
40 views

Deleting a contractible subspace is the same as deleting a point

Let $X$ be a topological space and $A$ and $B$ are subspaces of $X$. Suppose that $A$ is contractible. I know that taking the quotient does not affect the homotopy type, that is $X/A$ is homotopy ...
5
votes
1answer
166 views

$\mathbb{R}P^n$ is orientable iff $n$ is odd, without homology, without differential geometry

I am trying to prove that $\mathbb{R}P^n$ is orientable iff $n$ is odd. One way to do that is to calculate the homology of the space, and then use the (heavy?) theorem that states that a ...
0
votes
0answers
35 views

equivalent characterizations of manifolds such that configuration spaces are homotopy equivalent

Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th ...
2
votes
1answer
44 views

For which varieties is the natural map from the Chow ring to integral cohomology an injection?

For a smooth projective complex variety $X$ over $\mathbb{C}$, there is a natural map from its Chow ring $\mathbb{A}^*(X)$ into even integral cohomology $H^{2*}(X)$ of its (often implicitly ...
3
votes
1answer
43 views

Intersection form on $S^2 \tilde \times S^2$

The intersection form on the nontrivial $S^2$-bundle over $S^2$, denoted $S^2 \tilde \times S^2$, can be written as $\left[\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}\right]$ with ...
7
votes
0answers
78 views

Is it the case that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism?

So as the question statement asks, is it necessarily the case that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism? I suspect the answer is yes, but I don't know ...
1
vote
1answer
47 views

An intutive proof of 'replacing two-caps by a handle'

I am trying to understand a statement given in Polchinski Vol.1 - a torus with cross-cap can be obtained either as (g,b,c) = (0,0,3) or as (1,0,1), trading two cross-caps for a handle. Here, g is ...
4
votes
1answer
90 views

How to see that SL(2,C) is simply connected?

I started reading about Lie groups and right now I'm trying understand why $SL(2,\mathbb{C})$ is simply connected. I have shown that $SU(2)$, being diffeomorphic to $S^3$, is simply connected. So my ...
4
votes
0answers
58 views

Analogy between Galois groups and fundamental groups

I've heard that there is an analogy between algebraic field extensions and covers (in topology). In this analogy Galois extensions correspond to Galois covers and Galois groups correspond to ...
1
vote
1answer
35 views

Can this statement about winding number generalized?

Definition Let $\alpha$ be a path in $\mathbb{C}\setminus\{z_0\}$. Since $\mathbb{C}\rightarrow \mathbb{C}\setminus \{z_0\}:z\mapsto e^z$ is a covering map, $\alpha$ can be decomposed as ...
2
votes
1answer
98 views

The first Kirby move and $\mathbb{C}P^2$

A surgery on a link in $S^3$ can be regared as 2-hadnle attachements to $D^4$ (resulting 4-manifold $W$ whose boundary is the result of the surgery). I would like to know how the first Kirby move ...
2
votes
1answer
33 views

Difference between retract and deformation retract

I have a trouble with distinguishing retraction and deformation retraction intuitively. That is, deformation retraction is informally an operation on a space which continuously deform(for an ...
1
vote
0answers
27 views

Bundle map is isomorphism iff it covers a homeomorphism

Consider $P_0$ and $P_1$ principal G-bundles with projection maps $\pi_0, \pi_1$, respectively; $f:P_0 \rightarrow P_1$ a continuous G-equivariant map (i.e. a bundle map) and $g:X_0 \rightarrow X_1$ ...
0
votes
1answer
35 views

Mistake in Gabriel-Zisman regarding change-of-base of topological spaces?

In III.2.2 of Gabriel-Zisman, a Proposition is asserted which says that the base of change functor sending $X \to B$ to $X \times_{B} B'$, for any $B' \to B$ commutes with colimits in the $X$ ...
0
votes
0answers
50 views

$\Bbb R^5$ without two circles and line

How to prove that $\Bbb R^5$ without two $S^1$ and one $\Bbb R^1$ is homotopiclly equivalent to wedge of $(\Bbb R^5\backslash \Bbb R^1)$ and two copies of $(\Bbb R^5\backslash S^1)$? The only idea ...
1
vote
1answer
38 views

How do I prove that $R^n\setminus R^k$ is homeomorphic to $S^{n-k-1}\times R^{k+1}$?

Let $k,n$ be positive integers such that $k<n$. How do I prove that $\mathbb{R}^n\setminus \mathbb{R}^k$ is homeomorphic to $S^{n-k-1}\times \mathbb{R}^{k+1}$? I tried to put specific integers in ...
1
vote
2answers
49 views

Pullbacks and homotopy equivalences

Say I have a map between pullback squares $(Y \rightarrow Z \leftarrow X) \to (Y' \rightarrow Z' \leftarrow X')$. If the maps $X \to X'$, $Y \to Y'$ and $Z \to Z'$ are homotopy equivalences, does it ...
3
votes
0answers
53 views

Vector Bundles over Spheres

I would like to understand how to construct a vector bundle over the n sphere give a map of its equatorial $(n-1)$-sphere into the general linear group $GL_n(\mathbb{R})$. My thought is that one ...
0
votes
0answers
27 views

Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
5
votes
1answer
85 views

What's wrong in my thinking about Bézout's theorem?

First, I know that every hypersurface of degree $d$ defined in $\mathbb{CP}^n$ is diffeomorphic. By using this fact, I wanted to calculate the Euler characteristic of hypersurface of degree $d$. To ...
2
votes
0answers
75 views

Show that $\mathbb{Z}_4\rightarrow \mathbb{Z}_8\oplus \mathbb{Z}_2\rightarrow \mathbb{Z}_4$ is exact [duplicate]

I want to know whether $0\rightarrow \mathbb{Z}_4\stackrel{f}\rightarrow \mathbb{Z}_8\oplus \mathbb{Z}_2\rightarrow \mathbb{Z}_4\rightarrow 0$ is exact wrt group homomorphism under addition. Since ...
5
votes
1answer
53 views

Cofibration necessarily has closed image?

I know how to show that if $i: A \to X$ is a cofibration, then $i$ is injective, and in fact a homeomorphism onto its image. My question is, must the image necessarily be closed? I've tried ...
0
votes
0answers
23 views

Confused with notations about Leray's theorem for singular cohomology

The following theorem is copied from Bott's book Differential Forms in Algebraic Topology in Page 192: Theorem 15,11 {Leray's theorem for singular cohomology with coefficients in a commutative ...
2
votes
2answers
106 views

Hatcher Exercise 2.2.38

I'm struggling to show exactness at $C_n\oplus D_n$. Let's take $(x,y)\in C_n\oplus D_n$ in the kernel of $C_n\oplus D_n\to E_n$, i.e. the pushforwards $x', y'$ into $E_n$ resp. satisfy $x' + y' = ...
2
votes
1answer
41 views

Higher homotopy groups of wedge of circles.

Using van-kampen theorem, Fundamental group of wedge of n-circles is free group on n-generator. But I don't know how to calculate higher homotopy groups of wedge of spaces, in particular circles. I ...
4
votes
1answer
51 views

Fiber bundle with null-homotopic fiber inclusion

It is an exercise from Hatcher (exercise 31, page 392): For a fiber bundle $F \to E \xrightarrow{p} B$ such that the inclusion $F \hookrightarrow E$ is homotopic to a constant map, show that the long ...
14
votes
2answers
197 views

Geometric reason as to why $H^2$ of the Klein bottle is $\mathbb{Z}/2\mathbb{Z}$?

I was reading this document when I came across the following: Recall that $H^2(K; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$. Here $K$ denotes the Klein bottle. Is there a good geometric ...
4
votes
1answer
90 views

Fundamental group of open subsets of $\mathbb{R}^n$

Suppose that $U$ is an open subset of $\mathbb{R}^n$. What can be said about its fundamental group? I'm sure that the answer should be well known, since this is rather natural question.
2
votes
1answer
58 views

Orientability of Surfaces and the Fundamental Group

Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given ...