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

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80
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2k views

Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
61
votes
0answers
2k views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
33
votes
0answers
374 views

Does $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there. $\Bbb{CP}^{2n+1} \mathbin{\#} \Bbb{CP}^{2n+1}$ supports a complex structure: ...
15
votes
0answers
248 views

Definition of bordism - gluing manifolds with structure

In general, when you have a "cobordism category" (as defined in Stong's Notes on Cobordism Theory) you define two objects $M,N$ to be bordant when there exist $U,V$ such that $M \amalg \partial U ...
14
votes
0answers
764 views

When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
13
votes
0answers
198 views

A short question on shriek maps

This should be easy but I don't quite see it. Let $M^m, N^n, X^d$ be compact, connected and oriented smooth manifolds. Let also $f:M\rightarrow X$ and $g:N\rightarrow X$ be transverse smooth maps. ...
12
votes
0answers
148 views

Getting the most general form of Mayer-Vietoris from the axioms of homology

I'd like to derive the most general form of the Mayer-Vietoris sequence from the Eilenberg-Steenrod axioms for homology (in particular: I do not want to use the definition of $H_\ast(X)$ in terms of ...
11
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0answers
252 views

Show $\mathbb{CP^2/CP^1}$ is not a retract of $\mathbb{CP^4/CP^1}$.

So, I have shown that the natural projection $\pi\colon \mathbb{CP^n}\rightarrow \mathbb{CP^n/CP^k}$ induces a monomorphism $\pi^*\colon H^*(\mathbb{CP^n/CP^k},\mathbb Z)\rightarrow ...
11
votes
0answers
201 views

Why is the Mazur swindle named so?

Often results or techniques in mathematics are called 'theorems'. Sometimes they are called 'tricks'. In no other context have I seen a result called a 'swindle'. Is there a historical reason for this ...
11
votes
0answers
779 views

Mapping cylinder is a CW complex

If you read the question entirely, it is not a duplicate. The first time I asked the question, I already gave the link to the similar question and explained why the answer is not satisfying. If you ...
11
votes
0answers
236 views

How did Chern pictured the first Chern number?

The first Chern number $\cal C$ is known to be related to various physical objects. Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to ...
11
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0answers
1k views

Cohomology ring of Grassmannians

I'm reading a paper called An Additive Basis for the Cohomology of Real Grassmannians, which begins by making the following claim (paraphrasing): Let $w=1+w_1+ \ldots + w_m$ be the total ...
10
votes
0answers
180 views

Quotient Groups and Covering Spaces in Painting Hanging

Consider the $1$-out-of-$n$ painting hanging problem: Given $n$ nails in a wall, how can we hang a painting such that upon removal of any nail, it falls. This has a nice interpretation as a problem in ...
10
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0answers
199 views

Computing the homology class of a curve using Mayer-Vietoris

I am trying to compute the homology classes of various curves on a $6$-punctured torus $X$. I can easily see that the homology group is isomorphic to $\mathbb Z^7$ by letting $X=A\cup B$ where $A$ is ...
9
votes
0answers
138 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 ...
9
votes
0answers
150 views

Codimension $1$ Embedding into $\mathbb{R}^{n+1}$

I am trying to determine which homotopy types can be realized by $n$-manifolds that have codimension one embeddings into $\mathbb{R}^{n+1}$. Suppose I have $X^n \subset \mathbb{R}^{n+1}$ and an ...
9
votes
0answers
110 views

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 ...
9
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156 views

Prerequisites for studying Perelman's proof of the Geometrization Conjecture

I want to set a course toward understanding Perelman's proof of the Geometrization Conjecture. I realize this will be a lengthy undertaking, but hopefully only on the order of one to two years. I am ...
9
votes
0answers
142 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
9
votes
0answers
153 views

Algebraic methods to compute the cohomology ring of the complex topology of a variety?

Suppose $V$ is an affine (resp. projective) subvariety of the affine (resp. projective) space $\mathbb A_\mathbb C^n$ (resp. $\mathbb P_\mathbb C^n$) with vanishing ideal $I\subseteq\mathbb ...
9
votes
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107 views

How to prove this is a fibre bundle?

Let $M^{2n}$ be $2n$ dimensional toric manifold over a simple polytope $P^n$. Let $\pi : M^{2n} \longrightarrow P^n$ be the orbit map of the torus action. Let $F^k$ be a $k$ dimensional face of $P^n$. ...
9
votes
0answers
195 views

Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $ [M \mathbin\sharp N] \in H_n(M \mathbin\sharp N) $ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$ [M \mathbin\sharp N] = [M]+[N]$$ ...
9
votes
0answers
210 views

Need help on how to compute the fundamental group of a space.

I'm studying for an oral qualifying exam and going through various past exams I find on the interwebs, including this mock exam from the University of Bath. One of the questions seems like it should ...
9
votes
0answers
179 views

Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert ...
9
votes
0answers
308 views

De Rham Cohomology of $M \times \mathbb{S}^1$

Let $M$ be a closed (compact, without boundary) $m$-dimensional manifold. I want to prove that $H^{k+1}(M \times \mathbb{S}^1) = H^k(M) \oplus H^{k+1}(M)$. ($H^k$ is the $k$-th De Rham cohomology ...
9
votes
0answers
223 views

Equivalence of two definitions of Whitehead Torsion

In their book Lecture Notes in Algebraic Topology, Davis and Kirk define the torsion of an acyclic chain complex $C$ in the following way: Since $C$ is acyclic, there exists a simple chain complexes ...
9
votes
0answers
311 views

When are maps between topological manifolds automatically surjective?

Take an injective (continuous) map $T:\mathbb{S}_1\to\mathbb{S}_1$. It's an obvious fact (though I can only prove it with nontrivial facts about homotopy) that $T$ is automatically surjective. I have ...
9
votes
0answers
202 views

Lemma 6.3 of Milnor's Lectures on the h-cobordism theorem

Milnor's statement is: "Let $M^r$ and $N^s$ be sub-manifolds of $V^{r+s}$ which are all smooth, compact, oriented and without boundary. If $p$ is a point of $M^r$ contained in an $r$-cell $U$, ...
9
votes
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173 views

How to prove that my sum coincides with a sequence on the Online Encyclopedia of Integer Sequences?

I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be $$\small \dim \tilde{H}_t(X; {\mathbb{Z}}_2) = ...
8
votes
0answers
143 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
8
votes
0answers
304 views

The status of $\mathbb{R}$ in homotopy theory.

The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...
7
votes
0answers
116 views

cohomology ring of a configuration space

Consider the following configuration space of triples of points. $C = \left\lbrace (z_1,z_2,z_3) \in (\mathbb C^*)^3, z_1 \ne z_2, z_1 \ne z_3, z_2 \ne z_3\right\rbrace - \left\lbrace (z_1,z_2,z_3) ...
7
votes
0answers
119 views

Open questions in Topological K-Theory

I am interested in knowing about current research in the Topological K-Theory, especially its interactions with String Theory. About one and a half decade back, there were some papers by Physicists ...
7
votes
0answers
149 views

Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
7
votes
0answers
186 views

Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{2}$

Let $f,g:\mathbb{C}^3\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^3\to \mathbb{C}^3$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...
7
votes
0answers
143 views

Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective embeddings $f:K_1\rightarrow K_2, g:K_2\rightarrow ...
7
votes
0answers
191 views

Why the Steinberg idempotent is idempotent?

Consider the group $GL_n(\mathbb{F}_p)$. We have the following subgroups : -$\Sigma_n$ the symmetric group (permutation matrices) -$B_n$ the Borel subgroup (upper triangular matrices) -$U_n$ the ...
7
votes
0answers
141 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
7
votes
0answers
118 views

There does not exist a map $S^2\times S^2\to \mathbb{CP}^2$ with odd degree.

The following is a problem from a topology qualifying exam I am studying for: Show there does not exist a map $S^2\times S^2\to \mathbb{CP}^2$ with odd degree. I think I am doing something wrong, ...
7
votes
0answers
117 views

A theorem of Kan regarding fibrant replacement

Recall that there is an adjunction $$\mathrm{Sd} \dashv \mathrm{Ex} : \mathbf{sSet} \to \mathbf{sSet}$$ where $\mathrm{Sd} (\Delta^n)$ is the first barycentric subdivision of $\Delta^n$. There is a ...
7
votes
0answers
355 views

Trying to Understand Lefschetz Pencils

I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) better, tho I would appreciate insights on condition i), and in general. A Lefschetz pencil on a $4$-manifold $X$ is a ...
7
votes
0answers
162 views

Kunneth formula on product of spheres

Kunneth formula is $$ H^\ast (S^K;{\bf Z})\otimes_{{\bf Z}} H^\ast (S^M;{\bf Z}) =H^\ast (S^K\times S^M;{\bf Z}) =\wedge_{\bf Z} [a,b]$$ where $K=2k+1<M=2m+1$, $H^\ast (S^K;{\bf Z}) = ...
7
votes
0answers
188 views

Poincaré duality and intersection

Let's take $X$ and $Y$ K3 surfaces and $Z\subset X\times Y$ an algebraic cycle of dimension 2. I know that the Poincarè dual of $Z$, namely $[Z]$, is in $H^4(X\times Y,\mathbb{Z})$ and by Kunneth ...
7
votes
0answers
520 views

Fundamental group of the complement of Borromean rings

I got a homework asking me to show the fundamental group of the complement of the Borromean rings (say, $S^3\setminus B$ where $B$ is the Borromean rings) I know there should be three generators, but ...
7
votes
0answers
134 views

Continuous choice of basis for subspaces

Consider the flag variety (or flag manifold, depending on who you are) $V=\mathrm {Fl} (3,\mathbb C)$ of complete flags of subspaces of $\mathbb C^3$. That is, an element of M is a tuple (L , P) ...
7
votes
0answers
280 views

Weak Bott periodicity vs. strong Bott periodicity

Bruno Harris' proof (or I guess also Bott's original proof) of Bott periodicity (see here for instance) shows that there is a homotopy equivalence $h\colon\mathbb{Z}\times BU \rightarrow \Omega^2 ...
7
votes
0answers
349 views

Steenrod Algebra - Converting between Milnor to Serre-Cartan basis'

I have been studying the mod 2 Steenrod Algebra (denoted $\mathcal{A}$), using Mosher & Tangora. We have the Serre-Cartan (or Adem basis): Let $I = \{i_1,i_2,\ldots,i_n\}$ be a sequence of ...
7
votes
0answers
234 views

Ring structure in the Serre spectral sequence

I've tried to understand what's going on in Example 1.5 on page 27-28 in Hatcher's notes on spectral sequences. There is one part in the reasoning that I can't understand here. He writes down a table ...
6
votes
0answers
75 views

Generalization of invariance of domain

Suppose I say a topological space $X$ has the invariance of domain property iff $S \subset X$ is open as soon as $S$ is homeomorphic to an open set of $X$. We know that $\mathbb{R}^n$ has this ...
6
votes
0answers
78 views

Prop 12.8 in Bott & Tu

Yo! This proposition in Bott & Tu have been haunting me for a year or so since I always have to come back to this book for references. More precisely, the second equality in Proposition 12.8 in ...