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)

1
vote
0answers
6 views

tensor products of Hopf algebras

Let $H_1,\cdots, H_n$ be Hopf algebras over the field $\mathbb{Z}_p$, $p$ prime. Then the tensor product $$ \bigotimes_{i=1}^nH_i $$ is still an algebra over $\mathbb{Z}_p$. Is $ \otimes_{i=1}^nH_i $ ...
4
votes
1answer
50 views

Is “Categories and Sheaves” a good followup to Aluffi's “Algebra: Chapter 0”?

I'm about to finish Aluffi's "algebra: chapter 0" and am a bit confused as to what should be my next move. I've been planning to read Tom Dieck's Algebraic Topology for some time now. I glimpsed at it ...
11
votes
1answer
865 views

Does May's version of groupoid Seifert-van Kampen need path connectivity as a hypothesis?

May's A Concise Course in Algebraic Topology gives the following statement of the Seifert-van Kampen theorem for fundamental groupoids $\Pi(X)$ (section 2.7): Theorem (van Kampen). Let ...
10
votes
2answers
390 views

Ehresmann Connection of the tangential bundle & Chern classes

I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking. The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the ...
0
votes
2answers
55 views

Extending a homotopy equivalence

I have a basic question regarding homotopy equivalence. Let $X$, $Y$, and $Z$ be three subsets of $\mathbb{R}^2$ such that $(X\cap Z)\subset (Y\cap Z)$ are homotopy equivalent, and $X\setminus Z = ...
1
vote
1answer
52 views

What is a homology of chain complex?

First of all I don't know much about homological algebra, and algebraic topology I just took a class using Kinsey book (Topology of Surfaces) which is undergraduate math class and I learned what is ...
5
votes
1answer
57 views

What is the relation between cohomology of $K(\mathbf{Z}_2, n)$ and $K(\mathbf{Z}_2, n+1)$?

I'm reading a proof to a theorem, "Suppose $a, b: H^*(-, \mathbf{Z}_2)\to H^{*+k}(-, \mathbf{Z}_2)$ are two stable (commuting with suspension isomorphism) cohomology operation of degree k. If ...
0
votes
0answers
32 views

Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy ...
1
vote
1answer
44 views

Show $X$ is simply-connected given properties of two subsets

I'm given: $X$ is a manifold, $X = U\cup V$ for $U,V \subset X$ open, connected and simply-connected, with $U \cap V$ connected. And given this, I want to show $X$ is simply connected. Attempt I ...
3
votes
1answer
27 views

localization of the Pontrjagin ring of an $H$-space with respect to $\pi_0$

Let $X$ be an $H$-space. Let $F$ be a field. Then $H_*(X;F)$ is a Hopf algebra over $F$. According to group-like elements in the Hopf algebra of the homology of H-spaces, $$ \pi_0(X)=\{g\in ...
2
votes
0answers
23 views

Gysin sequence for the sphere bundle $B[O_a \times O_B]^+ \to BO_a \times BO_b$?

I have the sphere bundle $S^0 \hookrightarrow B[O_a \times O_b]^+ \stackrel{p}{\to} BO_a \times BO_b$ that can be thought of like: $B[O_a \times O_b]^+$ as the set of tuples of vector spaces $E^a$ ...
0
votes
1answer
78 views

What is a counterexample that a composition of covering maps not a covering map?

Let $p:X\rightarrow Y$ and $q:Y\rightarrow Z$ be covering maps. What would be an example that $q\circ p:X\rightarrow Z$ is not a covering map? I saw a counterexample here, but it was too complex. Is ...
15
votes
1answer
386 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 ...
4
votes
1answer
35 views

group-like elements in the Hopf algebra of the homology of $H$-spaces

Let $X$ be an $H$-space with product $\mu$. Then $H_*(X)$ is a Hopf algebra with product $\mu_*$. Let $\psi$ be the coproduct of the Hopf algebra $H_*(X)$. Define a subset $S$ of $H_*(X)$ as $$ ...
0
votes
0answers
50 views

What does $D^n$ refer to?

I'm not sure what object $D^n$ is in the following exercise: "Write down an explicit homeomorphism between $D^n/S^{n-1}$ and $S^n$." Thanks!
0
votes
0answers
22 views

Difference between $\mathbb{Z}$-acyclic and $\mathbb{Q}$-acyclic

In page 1853 of Graham's Handbook of Combinatorics, Volume 1, it is stated that for simplicial homology, $$\mathbb{Z_p}\textrm{-acyclic}\Leftrightarrow \mathbb{Z}\textrm{-acyclic} \Rightarrow ...
2
votes
1answer
49 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 ...
1
vote
0answers
18 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
43 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 ...
7
votes
1answer
92 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 ...
0
votes
1answer
26 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 ...
3
votes
1answer
148 views

Cross product of cohomology classes: intuition

Let $X$ and $Y$ be topological spaces and consider cohomology over a ring $R.$ Hatcher (in his standard Algebraic Topology text) defines the cross product of cohomology classes $$H^k(X) \times ...
0
votes
1answer
23 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
1answer
378 views

Computing the homology of $\mathbb{S}^2$ via Mayer-Vietoris

I'm trying to compute the homology of the $2$-sphere. I start by decomposing the sphere into a northern hemisphere and southern hemisphere, denoted by $A$ and $B$, respectively, and allow these two to ...
0
votes
0answers
33 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} ...
15
votes
1answer
205 views

Why would I define Alexander–Spanier cohomology?

I think I can motivate the definitions of simplicial, singular, de Rham, Čech, and sheaf (co)homology, more or less. I might want to understand bordism, and start by trying to understand ...
2
votes
1answer
62 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 ...
4
votes
0answers
41 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 ...
5
votes
3answers
109 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 ...
-1
votes
0answers
37 views

When is the Zariski closure of subset connected [on hold]

Let $R$ be a commutative ring and $\{P_i\}_{i\in I}$ be an arbitrary subset of $Spec(R)$ such that $\dfrac{R}{\bigcap_\limits{i\in I}{P_i}}$ is an indecomposable ring, how can we show that the ...
2
votes
1answer
15 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
1answer
74 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 ...
4
votes
1answer
48 views
7
votes
1answer
113 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 $x_1 x_2$, $x_3 x_4$, ...
4
votes
1answer
41 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
1answer
37 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".
2
votes
1answer
38 views

Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
5
votes
1answer
129 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 ...
7
votes
0answers
60 views
+50

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 ...
3
votes
0answers
28 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
67 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 ...
0
votes
0answers
32 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 ...
1
vote
2answers
186 views

Why cellular maps induce maps of chain complexes?

If $X$ is a CW-complex and $f:X\rightarrow X$ a cellular map. Then why it induces a map of chain complexes $f_*:C_*(X)\rightarrow C_*(X)$. (Why it commutes with the differential?).
2
votes
2answers
39 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 ...
4
votes
0answers
46 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 ...
1
vote
1answer
36 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 ...
0
votes
1answer
76 views

Questions on CW-complexes

I am trying to proof the following two statements. If $X_1 \subset \dots \subset X_i \subset \dots$ is a infinite sequence of CW-complexes, then $X = \bigcup X_i$ is a CW-complex and each $X_i$ is a ...
2
votes
1answer
39 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 ...
0
votes
0answers
32 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 ...