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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf  Topologia das Variedades \\[15mm]
\small Cohomology, lecture 10: Hopf theorem }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small\bf Misha Verbitsky  
\\[2mm]
June 12, 2018\\[4mm]

{\tiny\bf IMPA, Tuesdays and Thursdays, 10:30, Sala 224}
}
\end{center}

\newpage

{\bf\blue Bialgegras}

\definition
Let $A^\sbullet$, $B^\sbullet$ be graded commutative algebras
The {\bf \blue tensor product algebra} is $A^\sbullet\otimes B^\sbullet$
with the product $a \otimes b \cdot a' \otimes b' = (-1)^{\tilde b \tilde a'} aa' \otimes bb'$.

\remark
By K\"unneth formula, $H^\sbullet(X\times Y)$ is isomorphic to $H^\sbullet(X)\otimes H^\sbullet(Y)$
as an algebra.

\definition
Let $A^\sbullet$ be a graded commutative algebra over a field $k$.
We say that $A^\sbullet$ is a {\bf \blue bialgebra} if it is 
equipped with a homomorphism of algebras $A \stackrel \Delta \arrow A\otimes A$,
called {\bf \blue comultiplication} which is {\bf \blue coassociative},
that is, satisfies
\[ \Delta\circ \Delta\otimes \Id_A =  \Delta\circ \Id_A \otimes \Delta:\; 
A \arrow A\otimes_k A \otimes_k A.
\]
{\bf \blue Counit} of a bialgebra is an algebra homomorphism
 $A \stackrel \epsilon \arrow k$ which satisfies
$\Delta \circ  (\epsilon\otimes\Id_A)=\Delta \circ (\Id_A \otimes \epsilon)=\Id_A$
In the sequel, we shall tacitly assume that all bialgebras have counit.

\remark
{\bf \purple Coassociative comultiplication means that the dual space $(A^\sbullet)^*$
is equipped with an algebra structure.} Compatibility of comultiplication
with the multiplication in $A^\sbullet$ means that {\bf \purple this algebra structure
on $(A^\sbullet)^*$ is given by a morphism of $A$-modules.}


\newpage

{\bf \blue Examples of bialgebras}

\example
Let $N$ be a set equipped with an associative
operation $N \times N \stackrel m \arrow N$ with unit $e$
(such a structure is called {\bf \blue the structure of a 
monoid}, or {\bf \blue semigroup with unit}
Then {\bf \red the ring of $k$-valued functions $C(N)$ is a
bialgebra}, with comultiplication morphism given by
$m^*:\;  C(N) \arrow C(N\times N)= C(N)\otimes_k C(N)$,
and counit $\epsilon(v) = v(e)$.

\remark
The notion of a bialgebra is an abstraction of this
observation: heuristically speaking, {\bf \blue bialgebras
are alrebras of functions on monoids}.

\example
Let $N$ be a topological space equipped with a 
continuous map $N \times N \stackrel m \arrow N$
inducing the structure of a monoid. Consider the
comultiplication on the cohomology algebra
$H^\sbullet(N)$, given by 
 $m^*:\;  H^\sbullet(N) \arrow H^\sbullet(N\times N)= H^\sbullet(N)\otimes_k H^\sbullet(N)$.
{\bf \red Then $H^\sbullet(N)$ is a bialgebra}.
Indeed, coassociativity of $m^*$ follows from associativity of $N$,
and counit is given by the pullback to $H^\sbullet(e)=H^0(e)=k$.

\newpage

{\bf \blue $H$-spaces}

\definition
{\bf \blue An $H$-space} is a topological space
$M$ equipped with a continuous map $M \times M\stackrel \mu \arrow M$
(``the multiplication map'') and an element $e\in M$
(``the unit'') which satisfy ``semigroup conditions up to 
homotopy'', namely the following.

* {\bf \blue Homotopy associativity:} the maps 
$\mu\times \Id\circ mu:\; M\times M \times M \arrow M$
and  $\Id\times \mu\circ \mu:\; M\times M \times M \arrow M$ 
are homotopic.

* {\bf \blue Homotopy unit:} the map $\mu:\; M \times \{e\} \arrow M$
is homotopic to identity.

\example Clearly, {\bf \purple any toplogical group is an $H$-space.}

\claim
Let $M$ be an $H$-space. {\bf \red Then the cohomology algebra $H^\sbullet(M)$
is a bialgebra.}
\endproof

\newpage

{\bf \blue Loop spaces as $H$-spaces}

\example Let $\Omega(M, x)$ be the space of loops, that is, paths
$\gamma:\; [0, 1]\arrow M$ starting and
ending in $x$. We can multiply loops by mapping a pair $\gamma_1, \gamma_2:\; [0, 1]\arrow M $
to a loop $\gamma_1\gamma_2\; [0, 1]\arrow M$ 
equal to $\gamma_1(2t)$ on $[0, 1/2]$ and to $\gamma_2(2t-1)$ on $[1/2, 1]$.
The homotopy unit is the constant loop. This defines 
{\bf \red the structure of an $H$-group on the loop space}.

\remark The topology on the loop space $\Omega(M, x)$ can be defined,
for example, {\bf \purple by assuming that $M$ is a metric space, and
consider the uniform topology on the maps $\gamma:\; [0, 1]\arrow M$.}
In more generality, we take {\bf\blue the compact-open topology},
with the base sets consisting of all loops which map a given
compact $K\subset [0,1]$ to an open set $U\subset M$.

\newpage

{\bf \blue Bialgebras of finite type}

Let $V^\sbullet$ be a graded vector space.
Denote by $\Sym_{gr}(V^\sbullet)$
the tensor product $\Sym^*(V^\even)\otimes \Lambda^*(V^\odd)$
with a natural grading. On $\Sym^*(V^\even)\otimes \Lambda^*(V^\odd)$
one has a natural structure of an algebra.

\definition
{\bf \blue Free commutative algebra} is a polynomial algebra.
{\bf \blue Free graded commutative algebra} is
$Sym_{gr}(V^\sbullet)$, where $V^\sbullet$ is a graded vector space.

\definition
{\bf \blue A graded algebra of finite type} is an algebra
graded by $i \geq 0$, with all graded components finitely-dimensional.

\theorem {\bf \blue (Hopf theorem)}
Let $A^\sbullet$ be a graded bialgebra of finite type over a field
$k$ of characteristic 0. {\bf \red Then $A^\sbullet$ is a free graded 
commutative $k$-algebra}.

\remark
This allows one to compute the 
multiplicative structure {\bf \purple on all Lie groups
and on all loop spaces of finite CW-spaces.}

\remark For any Lie group $H^*(G)$ is finite-dimensional,
hence {\bf \red $H^*(G)$ is isomorphic to a Grassmann algebra.}
In particular, $\dim H^*(G)= 2^n$.


\newpage

{\bf \blue Heinz Hopf (1894-1971)}

\begin{center}
\epsfig{file=Hopf_3.jpeg,width=0.45\linewidth}\\

Heinz Hopf (1894-1971)
\end{center}


\newpage

{\bf \blue Primitive elements in a bialgebra}

\definition
An element $x$ of a bialgebra is called {\bf \blue primitive}
if $\Delta(x)= x\otimes 1 + 1\otimes x$.

\remark First, the Hopf theorem is proven for all
bialgebras, generated (multiplicatively) by the primitive
elements, and then we prove that finite type bialgebras
are generated by primitive elements.

\definition
Let $A$ be a Hopf algebra, and $P\subset A$ the
space of primitive elements. Consider the natural
homomorphism $\Sym_{gr}(P)\stackrel \psi\arrow A$.
We say that $A$ {\bf \blue is free up to degree $k$}
if $\bigoplus^{i \leq k} \Sym_{gr}^i(P)\stackrel \psi\arrow A$
is an embedding. 

\remark
The following lemma {\bf \purple immediately implies Hopf theorem
for all bialgebras generated by primitive elements.}

\lemma Let $A^\sbullet$ be a bialgebra which is free up to degree $k$.
{\bf \red Then $A^\sbullet$ is free up to degree $k+1$.}

\newpage

{\bf \blue Hopf theorem for bialgebras generated by primitive elements}

\lemma
 Let $A^\sbullet$ be a bialgebra which is free up to degree $k$.
{\bf \red Then $A^\sbullet$ is free up to degree $k+1$.}

\pstep Let $\{x_i\}$ be a basis in the space $P$ of primitive elements.
Consider a polynomial relation of degree $k+1$, say,
$Q(x_1, ..., x_n)=0$, and represent it as a polynomial of $x_1$
with coefficients which are polynomials of $x_2, ..., x_n$:
$Q= Q_m x_1^m+ Q_{m-1} x^{m-1}_1 + ... + Q_0$.
%Since $\psi:\; \bigoplus^{i \leq k} \Sym_{gr}^i(P)\stackrel \psi\arrow A$
%is an embedding, we have
Clearly, 
$\Delta(Q)= Q\otimes 1 + 1\otimes Q + R,$
where $R\in {\goth A}:= \left(\bigoplus^{i \leq k} \Sym_{gr}^i(P)\right)\otimes 
\left(\bigoplus^{i \leq k} \Sym_{gr}^i(P)\right)$. 

{\bf \green Step 2:} 
Since $\psi:\; \bigoplus^{i \leq k} \Sym_{gr}^i(P)\stackrel \psi\arrow A$
is an embedding, and elements of ${\goth A}$ belong to $\im \psi \otimes\im\psi$,
each element of ${\goth A}$ can be uniquely represented 
as a sum of monomials  $\lambda\otimes \mu$, where
$\lambda, \mu$ are degree $\leq k$ monomials on $x_i$. 
Denote by $\Pi:\; {\goth A} \arrow 
x_1 \otimes \left(\bigoplus^{i \leq k} \Sym_{gr}^i(P)\right)$
the projection to the sum of all monomials of form
$x_1\otimes \mu$. Since $\Delta(x_i) = x_i\otimes 1 + 1\otimes x_i$,
one has  $\Delta(x_1^m)= (x_1\otimes 1 + 1\otimes x_1)^m$,
{\bf \purple giving $\Pi(\Delta(x_1^m))= m x_1 \otimes x_1^{m-1}$.}

{\bf \green Step 3:} 
Let $\Pi(R):= x_1 \otimes R_0$.
Since $Q=0$ in $A$, its component $R_0$ is also equal to 0.
Then Step 2 gives
$0 = x_1 \otimes R_0 = x_1\otimes \sum_{i=1}^m m x_1^{m-1} Q_m$
where $Q_i$ are polynomials defined in Step 1.
Then all $Q_i=0$. \endproof

\newpage

{\bf \blue Algebras with filtration}

\remark Step 3 of the proof of previous lemma uses $\Char k=0$.
{\bf \red Hopf theorem is false for $\Char k >0$.}


\definition {\bf \blue Filtration} on an algebra $A$
is a sequence of subspaces $A_0 \supset A_1\supset A_2 \supset ...$
such that $A_i \cdot A_i \subset A_{i+j}$

\example
Let $I\subset A$ be an ideal. {\bf \blue the $I$-adic filtration}
is the filtration by the degrees of the ideal $I$:
$A \supset I \supset I^2 \supset I^3 \supset ...$.

\definition
Let $A_0 \supset A_1\supset A_2 \supset ...$ be a filtered algebra.
{\bf\blue The associated graded algebra} is $A_{gr}:=\bigoplus_i A_i/A_{i+1}$.

\lemma
Let $A \supset I \supset I^2 \supset I^3 \supset ...$ be an adic filtration,
and $A_{gr}:=\bigoplus_i I^i/I^{i+1}$ the associated graded algebra.
{\bf \red Then $A_{gr}$ is generated by its first and second graded components
$A/I \oplus I/I^2$}.

\proof Indeed, $I^k/I^{k+1}$ is generated by products of $k$ elements in 
$(I/I^2)$. \endproof 

\newpage

{\bf \blue Proof of Hopf theorem}

\definition
{\bf \blue Augmentation ideal} in a bialgebra 
is the kernel of the counit homomorphism
$\epsilon:\; A \arrow k$.

\remark The counit condition gives
$x= [\epsilon \otimes \Id_A](\Delta(x))$ and
$x=  [\Id_A\otimes \epsilon](\Delta(x))$, hence
\[ \Delta(x) = 1\otimes x + x\otimes 1 \mod (Z\otimes Z)
\]
\!\!\!\!\!\!\!\theorem {\bf \blue (Hopf theorem)}
Let $A$ be a graded bialgebra of finite type over a field
$k$ of characteristic 0. {\bf \red Then $A$ is a free graded commutative $k$-algebra}.

\pstep Consider the filtration of $A$ by the degrees of the augmentation ideal $A$,
and let $A_{gr}:=\bigoplus_i Z^i/Z^{i+1}$ be the associated graded algebra.
{\bf \purple Since $\Delta(x) = 1\otimes x + x\otimes 1 \mod (Z\otimes Z)$, one has
$\Delta(Z^p) \subset \bigoplus_{i+j=p} Z^i \otimes Z^j$.}

{\bf \green Step 2:} This implies that all bialgebra operations on $A$
are compatible with the $Z$-adic filtration. 
{\bf \purple Therefore, $A_{gr}$ is also a Hopf algebra}. 

{\bf \green Step 3:} {\bf \purple The algebra $A_{gr}$ is multiplicatively
generated by $Z^1/Z^2$}. Since $\Delta(x) = 1\otimes x + x\otimes 1 \mod (Z\otimes Z)$,
all elements of $Z^1/Z^2$ are primitive in $A_{gr}$. Therefore,
the algebra $A_{gr}$ is generated by primitive elements.
This implies that {\bf \purple  $A_{gr}$ is a free algebra generated by its space of
primitive elements.}

\newpage

{\bf \blue Proof of Hopf theorem (2)}

{\bf \green Step 4:} Let $x_i$ be a basis in the space of
primitive elements of $A_{gr}$, and let $\tilde x_i$ be
a representative of each of $x_i\in Z^k/Z^{k+1}$ in $Z_k$,
of the same parity as $x_i$. {\bf \purple Since there is no non-trivial
relations between $x_i$, there are no non-trivial relations
between $\tilde x_i$. } It remains to show that
$\tilde x_i$ generate $A$.

{\bf \green Step 5:} Return to the grading originally
given on $A$. Since $\epsilon$ is compatible with grading,
the ideal $Z$ is a direct sum of its graded components,
and the algebra $A_{gr}$ is equipped with a grading induced from $A$.
Dimensions of the graded components $A^p$ and $A_{gr}^p$
of $A$ and $A_{gr}$ are equal,
because any filtered space is isomorphic as a vector space
to its associated graded space. Let $\{y_i\}$ be a set of
monomials of $x_i\in A_{br}$ giving a basis in the graded
component $A^p_{gr}$, and $\{\tilde y_i\}$ the corresponding
monomials in $A^p$. {\bf \purple Since $\{y_i\}$ are linearly independent,
the monomials $\{\tilde y_i\}$ are linealry independent,
and since $\dim A^p= \dim A_{gr}^p$, these monomials
generate $A^p$.} We have shown that $A^\sbullet$ is freely
generated by the vectors $\{\tilde y_i\}$.
\endproof




\end{document}
