algebraic topology – Equivalence relation in singular homology

I’m currently reading Rotman’s Introduction to algebraic topology, and I’m struggling to understand what singular homology is.

The definition of a singular simplex isn’t clear to me. If a singular simplex is any continuous map from the standard n simplex, then there are uncountably many of them (except for some special spaces). The point that I don’t get is how all of those simplexes “cancel” each other in the homology group. I read about an equivalence relation that takes care of it but I didn’t understand.

Thanks in advance!

algebraic topology – Homology of Torus with two discs glued on the inside

I am trying to figure out the homology of the space $X$, where $X$ is a $T^2$ torus with two discs $D_1,D_2$ glued on the inside is (see drawing).

What is the CW-complex structure of $X$?

I thought I would take one 0-cell $p$ and attach four 1-cells $a,b,alpha,beta$ where $a,b$ correspond to the 1-cells of $T^2$ in $X$. Then I would attach three 2-cells $A,D_1,D_2$, where $A$ is attached via the identification of the torus and $D_1,D_2$ are attached to $alpha,beta$ respectively. So we get the chain complex $0tomathbb{Z}^3tomathbb{Z}^4tomathbb{Z}to0.$ Is that the right approach?

What are the boundry maps on the chain complex?

My try: The differentials $d_0,d_1$ are both zero as there is only one 0-cell. The Space is path connected thus $H_0(X)=mathbb{Z}$. But what about $d_2: mathbb{Z}^3tomathbb{Z}^4$? Is there a problem with the approach?

The space in question

cohomology – Optimality condition of the harmonic form representatives of a homology class

In “Hodge theory on metric spaces, Smale et al.” the $d$-th harmonic forms of the Hodge Laplacian $Delta_d=delta^* delta+delta delta^*$ satisfying $Delta_d(f)=0$ are claimed to be special representatives of their homology class, interpreted as satisfying an optimality condition.

Which is this optimality condition?

homology cohomology – Why are Simplicial Chains assumed Constant?

Just curious and wondering if I am missing something obvious:

Let K be a simplicial complex. Why are Simplicial chains with coefficients in a group G described as $$ g sigma_i ^m $$

for g in G and $ sigma_i^m $ an m-simplex in the m-th chain group $C_m(K,G)$, i.e., chains seem

to be assumed to be constant in every simplex?

algebraic topology – Let $f,g:S^5to S^5$ are degree 2 maps. Then show that we can find a map $psi:C(gcirc f)to C(g)$ and find its effect on homology groups.

Here $C(f)=Cone(f:Xto Y)$ is the pushout of
$require{AMScd}$
begin{CD}
X @>>>C(X)\
@V{f}VV @VVV\
Y @>>> C(f)
end{CD}

And $C(X):=frac{Xtimes(0,1)}{Xtimes 1}$ denotes the cone of $X$.

For notational convenience we denote $f:S^5_ato S^5_b$ and $g:S^5_bto S^5_c$. Then we know

$$C(gcirc f)=frac{S^5_csqcup C(S^5_a)}{gcirc f(x)sim ((x,0))}$$

First we define the map $displaystyle{phi:S^5_csqcup C(S^5_a)to C(g)=frac{S^5_csqcup C(S^5_b)}{g(x)sim ((x,0))}}$ by
$$phi(x)=(x) forall xin S^5_c$$ and $$phi((x,t))=(f(x),t) forall ((x,t))in C(S^5_a)$$

Then $phi(gcirc f(x))=((f(x),0))=phi(((x,0)))$. Hence, by the universal property of quotient spaces $phi$ induces the map
$$psi:C(gcirc f)to C(g)$$
Now I have to calvculate the $psi_*:H_k(C(gcirc f))to H_k(C(g))$.

I know that as $f,g$ has degree $2$, so $gcirc f$ has $4$. Hence, $$C(gcirc f)cong M(Bbb{Z}/4,5)text{ and } C(g)cong M(Bbb{Z}/2,5)$$
where $H_5(M(Bbb{Z}/n,k))=Bbb{Z}/n, H_0(M(Bbb{Z}/n,k))=Bbb{Z}$ and $0$ otherwise.

Can anyone help me to compute $psi_*:H_k(C(gcirc f))to H_k(C(g))$ for $k=0,5$? Thanks for help in advance.

at.algebraic topology – Induced map in homology for a map to a loop space

Suppose $Y$ is an $(n-1)$-connected space, $n>2$, so we have Hurewicz isomorphisms $pi_n(Y)cong H_n(Y)$ and $pi_{n-1}(Omega Y)cong H_{n-1}(Omega Y)$. Let a map $alphacolon XtoOmega Y$ be given. Naturally it induces a map $betacolon Xtimes S^1to Y$. I want to show the following diagram is commutative:
$$require{AMScd}
begin{CD}
H_{n-1}(X) @>times(S^1)>> H_n(Xtimes S^1)\
@Valpha_*VV @Vbeta_*VV \
H_{n-1}(Omega Y) @<cong<< H_n(Y).
end{CD}
$$

Since $Omega Y$ is path connected, we may homotope $alpha$ to a based map. Then $beta$ factors though the reduced suspension $Sigma X$. If $X=S^{n-1}$ is a sphere, the commutativity would then follow from tracking down the definition of $pi_n(Y)xrightarrow{cong}pi_{n-1}(Omega Y)$. However I don’t know how this helps for the general case.

One can also phrase the question in cohomology in the obvious way. (In particular the cross product $times(S^1)$ will be replaced by the slant product $/(S^1)$.)

at.algebraic topology – Homology of a fiber as a cotorsion product

Let $K$ be a field. For any differentially graded coalgebra $A$ over $K$, any differentially graded right $A$-comodule $M$ over $K$ and any differentially graded left $A$-comodule $N$ over $K$ let
$mathrm{Cotor}_A(M,N)$ denote the cotorsion product of $M$ and $N$ relative to $A$.

The graded $K$-vector space $mathrm{Cotor}_A(M,N)$ is by definition the homology of the totalization of the cosimplicial cochain complex over $K$ with $n$-th term $M otimes A^{otimes n} otimes N$,
where the tensor product is in cochain complexes over $K.$

Let $X to Y$ be a Serre fibration between connected spaces and $F$ its fiber over a given point $y$ of $Y.$

If $Y$ is simply connected, by a theorem of Eilenberg and Moore there is a canonical isomorphism
begin{equation}
H_*(F;K)cong mathrm{Cotor}_{C_*(Y; K)}(C_*(X; K),C_*(*; K)), (**)
end{equation}

where $C_*(-;K)$ are singular chains with coefficients in the field $K.$

Can we replace the condition that $Y$ is simply connected by a weaker condition?

For example, is there still a canonical isomorphism $(**)$ if $Y = BG = K(G,1) $ for $G$ a derived p-complete abelian group?

linear algebra – Why the diagonal elements of the Smith normal form of a boundary matrix are the torsion coefficients of a homology module?

Can you help me in proving/justifying the isomorphism going between $H_p$ and $left(bigoplus_{i=1}^{l_{p}} R / b_{p i} Rright)$.

Where $H_p$ is the p-th homolgy module, R is a commutative PID, and $b_{pi}$‘s are the diagonal elements in the Smith normal form of the boundary matrix.

Please support in this. Any hints would be highly appreciated!!

P.S. From the generalized version of the structure Theorem, $H_p$ is uniquely decomposed into:
$H_{p} simeq R^{beta_p} oplusleft(bigoplus_{i=1}^{l_{p}} R / b_{p i} Rright)$, $beta_p$ is the p-th Betti number (the rank) of $H_p$.

rt.representation theory – Irreducible representations of the symmetric group on homology of simplicial complex

I am following Wall’s paper A note on symmetry of singularities and I have some questions regarding representation theory and the homology of some objects:

Consider an action of $Sigma_k$ on a finite simplicial complex $X$ of dimension $n$, such that the action is simplicial and a simplex is fixed iff it is point-wise fixed. Consider $sigmainSigma_k$, then
$$chi_{Top}(X^sigma)=chi_{Sigma_k}(X)(sigma):= sum_i(-1)^i text{trace } sigma^*:H_i(X)rightarrow H_i(X), $$
where $X^sigma$ is the fixed subcomplex fixed by $sigma$ and $chi_{Top}$ the Euler-Poincaré characteristic.

My questions are:

1.- Is the sum of characters $chi_{Sigma_k}(X)$ called the equivariant Euler characteristic? Because I believe the equivariant Euler characteristic is the Euler characteristic of $H_*(X)^G$ in other texts.

2.- Can one say anything about the character on a single homology group instead of the alternated sum of all the groups? For example, compare the trace of $sigma$ on one $H_i$ and $H_i(X^sigma)$. I have a lot of troubles trying to isolate one homology group from this equality (with the extra structure that I have in my problem).

3.- Can one say something about the isotypes corresponding to a single irreducible representation of $Sigma_k$? For example, compare the number of copies of the $sign$ representations and $H_*(X^sigma)$ for some $sigma$.

If it is useful, in my problem I know that every $X^sigma$ has lower dimension than $X$, if $sigmaneq 1_{Sigma_k}$.

symplectic topology – Properties of $I_{mu}$ for Lagrangian Floer Homology in the Cotangent bundle

Following the notation of the book “Lagrangian intersection Floer theory anomaly and obstruction” suppose we have that our symplectic manifold is a cotangent bundle $T^*M$ with the canonical symplectic form and our two Lagrangian subamifolds are $L_0=phi^1(T^*_{p}M)$ and $L_1=T_q^*M$ for some $p,qin M$ and $phi^t$ is an hamiltonian flow.

Now if we consider the equivalence relation that give us the Novikov Covering where we have elements of the form $(l_p,w) $ and $(l_q,w’)$, I am interested in seeing at what happens to it. That is consider the homomorphisms $I_{omega}$ and $I_{mu}$. It is easy to see that due to the fact the cotangent bundle is an exact symplectic manifold we will have that $I_{omega}(C)=0$ for any $C=bar w# w’$ and so half of the equivalence relation is always true for any elements $(l_p,w)$ and $(l_p,w’)$ that I consider. Now I am interest on seeing what happens with the other one part of the equivalence relation.

So my question is if anybody knows or has any suggestion on checking wether $I_{mu}(C)=0$? Or more generally if our triple $(M,L_0,L_1)$ will be monotone , i.e, if there exists a $lambda >0$ such that $I_{omega}=lambda I_{mu}$?

(I have tried to look for references of these but I could´t find anything). Any help is appreciated. Thanks in advance.