Intersection form
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Contents |
1 Introduction
Let be a closed oriented
-manifold (PL or smooth). After Poincaré one studies the intersection number of transverse submanifolds or chains in
.
The intersection number gives a bilinear intersection product
![\displaystyle I_N=\cap_N=\cdot_N=\lambda_N\colon H_k(N;\Zz) \times H_{n-k}(N;\Zz) \to \Zz](/images/math/c/6/5/c65e912eb0870879e5df1afa6131e845.png)
defined on the homology of .
For
this is the intersection form of
.
For
the signature of this form is the signature
of
.
The intersection product is related to characteristic classes.
These are important invariants used in the classification of manifolds.
2 Definition
Tex syntax errorof
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Denote by the dual cell subdivision.
![[x]\in H_k(N;\Zz_2)](/images/math/7/0/7/707dad1546f3e2d2b347d942a382cab6.png)
![[y]\in H_{n-k}(N;\Zz_2)](/images/math/4/6/7/4679c012b7ce2e40274b6c40ad1762d2.png)
![x](/images/math/8/7/2/8725029ea89712eed8670bae64d30e47.png)
![y](/images/math/3/6/a/36a4dc9ccf2bdc09d800556724231fc6.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax errorand
![(n-k)](/images/math/f/8/a/f8a3098ddcbf5cb2e3a80741378b1736.png)
![T^*](/images/math/a/5/f/a5f159feaf5e6825f500308ae09f6b87.png)
Define the modulo 2 intersection number by the formula
![\displaystyle I_{N,2}(x,y)=x\cap_{N,2} y=\langle\, x \, , \, y\, \rangle := |x\cap y|\mod2\in\Zz_2.](/images/math/0/e/b/0eb766392569b92d16ec516ba2c74be4.png)
Define the modulo 2 intersection product
![\displaystyle I_{N,2}=\cap_{N,2}: H_k(N;\Zz_2) \times H_{n-k}(N;\Zz_2) \to \Zz_2\quad\text{by}\quad I_{N,2}([x],[y]):=I_{N,2}(x,y).](/images/math/3/a/9/3a9d69a54208e265e76fa92de305c3ef.png)
This product is well-defined because the intersection of a cycle and a boundary consists of an even number of points (by definition of a cycle and a boundary).
Analogously, counting intersections with signs, one defines the intersection number
![\displaystyle I_N(x,y)=x\cdot y=\langle\, x \, , \, y\, \rangle \in \Z](/images/math/7/e/8/7e8ae6e5eda55546f98444687fffc479.png)
of integer chains and
.
Clearly, the product of a cycle and a boundary is zero.
Hence this defines the above intersection product
.
Using the notion of transversality, one can give an equivalent (and `more general') definition as follows. Take a -chain
and an
-chain
which are transverse to each other. The signed count of the intersections between
and
gives the intersection number
. A particular case is intersection number of immersions.
Define the intersection product by
.
See [Kirby1989, Chapter II], [Skopenkov2005, Remark 2.3], [Skopenkov2015b, 6].
Using the notion of cup product, one can give a dual (and so an equivalent) definition:
![\displaystyle I_N(x,y) = \langle x^*\smile y^*,[N]\rangle \in \Z,](/images/math/b/0/9/b098fae44db83b9fa8ee187a581e1526.png)
where ,
are the Poincaré duals of
,
, and
is the fundamental class of the manifold
. We can also define the cohomology intersection product
![\displaystyle I_N^*: H^k(N;\Zz) \times H^{n-k}(N;\Zz) \to \Zz \quad\text{by}\quad I_N^*(p,q) = \langle p \smile q , [N] \rangle .](/images/math/d/d/3/dd301a192d5a20a4267f2bf7881bcfb7.png)
The definition of a cup product is `dual' (and so is analogous) to the above definition of the intersection product on homology, but is more abstract. However, the definition of a cup product generalizes to complexes (and so to topological manifolds). This is an advantage for mathematicians who are interested in complexes and topological manifolds (not only in PL and smooth manifolds).
If , the intersection product is the intersection form denoted by
.
A closely related important notion is linking form.
3 Properties
The following properties are easy to check using the simple direct definition; they also follow from simple properties of the cup product.
The intersection product is bilinear. Hence it vanishes on torsion elements. Thus it descends to a bilinear pairing
![\displaystyle H_k(N;\Zz) / \text{Torsion}\times H_{n-k}(N;\Zz) / \text{Torsion}\to\Zz.](/images/math/0/5/9/059a51bc10e313685bc0114254486825.png)
on the free modules. This pairing is unimodular (in particular non-degenerate) by Poincaré duality.
We have
![\displaystyle I_N(x,y) = (-1)^{k(n-k)}I_N(y,x).](/images/math/c/c/0/cc0f17a1775a32e39bb0d9b60a085052.png)
Hence for
- If
is even the form
is symmetric:
.
- If
is odd the form
is skew-symmetric:
.
4 Definition of signature
Let be a symmetric bilinear form on a free
-module.
Denote by
(
) the number of positive (negative) eigenvalues.
Note that since is symmetric, it is diagonalisable over the real numbers, so
(
) is the dimension of a maximal subspace on which the form is positive (negative) definite.
Then the signature of is defined to be
Tex syntax error
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
Tex syntax erroris defined to be the signature of the intersection form of
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
5 Equivalence of bilinear forms
Let and
be unimodular bilinear forms on underlying free
-modules
and
respectively. The forms
and
are called equivalent or isomorphic if there is an isomorphism
such that
.
The rank of is the rank of the underlying
-module
.
6 Skew-symmetric bilinear forms
The skew-symmetric hyperbolic form of rank ,
, is defined by the following intersection matrix
![\displaystyle \left( \begin{array}{cc} ~0 & ~1 \\ -1 & ~0 \end{array} \right) .](/images/math/f/3/a/f3ac0fdac865b4d550e21dcab0de6b1d.png)
Proposition 6.1.
Every skey-symmetri uni-modular bilinear form over ,
, isomorphic to the sum of some number of hyperbolic forms:
![\displaystyle q \cong \oplus_{i=1}^r H_-(\Zz).](/images/math/0/c/a/0ca6433f06c963c2693431c53b38e435.png)
In particular the rank of , in this case
, is even.
7 Symmetric bilinear forms
The classification of uni-modular definite symmetric bilinear forms is a deep and difficult problem. However the situation becomes much easier when the form is indefinite. Fundamental invariants are rank, signature and the following two.
A form is called definite if it is positive or negative definite, otherwise it is called indefinite.
A form may have two different types. It is of type even if
is an even number for any element
. Equivalently, if
is written as a square matrix in a basis, it is even if the elements on the diagonal are all even. Otherwise,
is said of type odd.
7.1 Classification of indefinite forms
There is a simple classification result of indefinite forms [Serre1970],[Milnor&Husemoller1973]:
![q, q'](/images/math/0/0/8/00867c957e16284b9243067ea7635467.png)
![\mathbb{Z}](/images/math/9/3/4/9349f2f6b15e7d4ce91c301ddd64bc06.png)
![q](/images/math/e/b/6/eb6af5b4e510c3c874d7d1f51d72393a.png)
![q'](/images/math/6/f/d/6fdb1bd0ad5208d967584a84fc65b257.png)
There is a further invariant of a unimodular symmetric bilinear form on
: An element
is called a characteristic vector of the form if one has
![\displaystyle q(c,x) \equiv q(x,x) \ (\text{mod} \ 2)](/images/math/a/7/4/a74ad4537107ba2c3a93c27b23682c57.png)
for all elements . Characteristic vectors always exist. In fact, when reduced modulo 2, the map
is linear. By unimodularity there therefore exists an element
such that the map
equals this linear map.
The form is even if and only if
is a characteristic vector.
If
and
are characteristic vectors for
, then there is an element
with
. This follows from unimodularity. As a consequence, the number
is independent of the chosen characteristic vector
modulo 8. One can be more specific:
Proposition 7.2.
For a characteristic vector of the unimodular symmetric bilinear form
one has
![\displaystyle q(c,c) \equiv \text{sign}(q) \ (\text{mod} \ 8)](/images/math/a/2/f/a2fb4177a192383dae312383b7f801c0.png)
Proof:
Suppose is a characteristic vector of
. Then
is a characteristic vector of the form
![\displaystyle q' = q \oplus \begin{pmatrix} 1 & \ 0 \\ 0 & -1 \end{pmatrix},](/images/math/4/3/a/43a487a575603d4bc0d597fc65ae01f3.png)
where form basis elements of the additional
summand with square
. We notice that
![\displaystyle q(c,c) = q'(c+e_+ + e_-, c+e_+ + e_-) .](/images/math/8/1/a/81a26702a8c96dd149c5023026f12b6c.png)
However, the form is indefinite, so the above classification theorem applies. In particular,
is odd and has the same signature as
, so it is equivalent to the diagonal form with
summands of (+1) and
summands of
. This diagonal form has a characteristic vector
that is simply a sum of basis elements in which the form is diagonal. Of course
. The claim now follows from the fact that the square of a characteristic vector is independent of the chosen characteristic vector modulo 8.
Corollary 7.3. The signature of an even (definite or indefinite) form is divisible by 8.
7.2 Examples, Realisations of indefinite forms
We shall show that any indefinite form permitted by the above theorem and corollary can be realised.
All possible values of rank and signature of odd forms are realised by direct sums of the forms of rank 1,
![\displaystyle b^+ (+1) \oplus b^- (-1) .](/images/math/d/1/b/d1bc34c5d3fda6e7a65a01c93b65e5f5.png)
An even positive definite form of rank 8 is given by the matrix
![\displaystyle E_8 = \left( \begin{array}{c c c c c c c c} \ 2 \ & \ 1\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ \\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 \\ \end{array} \right) .](/images/math/a/b/d/abd2d8d6026c5ebb8731ece7dcecc28c.png)
Likewise, the matrix represents a negative definite even form of rank 8.
Tex syntax errorgiven by
![\displaystyle H = \begin{pmatrix} \ 0 \ & \ 1 \ \\ 1 & 0 \end{pmatrix}](/images/math/3/c/5/3c5111cc3e76964de1e2d334965b144f.png)
determines an indefinite even form of rank 2 and signature 0. It is easy to see that the direct sums
![\displaystyle k E_8 \oplus l H](/images/math/3/9/4/394f2f02da4ec0f3793c0495ea6dc3a5.png)
with realise all unimodular symmetric indefinite even forms that are allowed by the above classification result. Here we use the convention that
is the
-fold direct sum of
for positive
and
is the
-fold direct sum of the negative definite form
.
8 References
- [Kirby1989] R.C. Kirby, The topology of 4-manifolds, Lecture Notes in Math. 1374, Springer-Verlag, 1989. MR1001966 (90j:57012)
- [Milnor&Husemoller1973] J. Milnor and D. Husemoller, Symmetric bilinear forms, Springer-Verlag, New York, 1973. MR0506372 (58 #22129) Zbl 0292.10016
- [Serre1970] J. Serre, Cours d'arithmétique, Presses Universitaires de France, Paris, 1970. MR0255476 (41 #138) Zbl 0432.10001
- [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
- [Skopenkov2015b] A. Skopenkov, Algebraic Topology From Geometric Viewpoint (in Russian), MCCME, Moscow, 2015, 2020. Accepted for English translation by `Moscow Lecture Notes' series of Springer. Preprint of a part
9 External links
- The Wikipedia page on Poincaré duality
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\displaystyle I_N=\cap_N=\cdot_N=\lambda_N\colon H_k(N;\Zz) \times H_{n-k}(N;\Zz) \to \Zz](/images/math/c/6/5/c65e912eb0870879e5df1afa6131e845.png)
defined on the homology of .
For
this is the intersection form of
.
For
the signature of this form is the signature
of
.
The intersection product is related to characteristic classes.
These are important invariants used in the classification of manifolds.
2 Definition
Tex syntax errorof
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Denote by the dual cell subdivision.
![[x]\in H_k(N;\Zz_2)](/images/math/7/0/7/707dad1546f3e2d2b347d942a382cab6.png)
![[y]\in H_{n-k}(N;\Zz_2)](/images/math/4/6/7/4679c012b7ce2e40274b6c40ad1762d2.png)
![x](/images/math/8/7/2/8725029ea89712eed8670bae64d30e47.png)
![y](/images/math/3/6/a/36a4dc9ccf2bdc09d800556724231fc6.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax errorand
![(n-k)](/images/math/f/8/a/f8a3098ddcbf5cb2e3a80741378b1736.png)
![T^*](/images/math/a/5/f/a5f159feaf5e6825f500308ae09f6b87.png)
Define the modulo 2 intersection number by the formula
![\displaystyle I_{N,2}(x,y)=x\cap_{N,2} y=\langle\, x \, , \, y\, \rangle := |x\cap y|\mod2\in\Zz_2.](/images/math/0/e/b/0eb766392569b92d16ec516ba2c74be4.png)
Define the modulo 2 intersection product
![\displaystyle I_{N,2}=\cap_{N,2}: H_k(N;\Zz_2) \times H_{n-k}(N;\Zz_2) \to \Zz_2\quad\text{by}\quad I_{N,2}([x],[y]):=I_{N,2}(x,y).](/images/math/3/a/9/3a9d69a54208e265e76fa92de305c3ef.png)
This product is well-defined because the intersection of a cycle and a boundary consists of an even number of points (by definition of a cycle and a boundary).
Analogously, counting intersections with signs, one defines the intersection number
![\displaystyle I_N(x,y)=x\cdot y=\langle\, x \, , \, y\, \rangle \in \Z](/images/math/7/e/8/7e8ae6e5eda55546f98444687fffc479.png)
of integer chains and
.
Clearly, the product of a cycle and a boundary is zero.
Hence this defines the above intersection product
.
Using the notion of transversality, one can give an equivalent (and `more general') definition as follows. Take a -chain
and an
-chain
which are transverse to each other. The signed count of the intersections between
and
gives the intersection number
. A particular case is intersection number of immersions.
Define the intersection product by
.
See [Kirby1989, Chapter II], [Skopenkov2005, Remark 2.3], [Skopenkov2015b, 6].
Using the notion of cup product, one can give a dual (and so an equivalent) definition:
![\displaystyle I_N(x,y) = \langle x^*\smile y^*,[N]\rangle \in \Z,](/images/math/b/0/9/b098fae44db83b9fa8ee187a581e1526.png)
where ,
are the Poincaré duals of
,
, and
is the fundamental class of the manifold
. We can also define the cohomology intersection product
![\displaystyle I_N^*: H^k(N;\Zz) \times H^{n-k}(N;\Zz) \to \Zz \quad\text{by}\quad I_N^*(p,q) = \langle p \smile q , [N] \rangle .](/images/math/d/d/3/dd301a192d5a20a4267f2bf7881bcfb7.png)
The definition of a cup product is `dual' (and so is analogous) to the above definition of the intersection product on homology, but is more abstract. However, the definition of a cup product generalizes to complexes (and so to topological manifolds). This is an advantage for mathematicians who are interested in complexes and topological manifolds (not only in PL and smooth manifolds).
If , the intersection product is the intersection form denoted by
.
A closely related important notion is linking form.
3 Properties
The following properties are easy to check using the simple direct definition; they also follow from simple properties of the cup product.
The intersection product is bilinear. Hence it vanishes on torsion elements. Thus it descends to a bilinear pairing
![\displaystyle H_k(N;\Zz) / \text{Torsion}\times H_{n-k}(N;\Zz) / \text{Torsion}\to\Zz.](/images/math/0/5/9/059a51bc10e313685bc0114254486825.png)
on the free modules. This pairing is unimodular (in particular non-degenerate) by Poincaré duality.
We have
![\displaystyle I_N(x,y) = (-1)^{k(n-k)}I_N(y,x).](/images/math/c/c/0/cc0f17a1775a32e39bb0d9b60a085052.png)
Hence for
- If
is even the form
is symmetric:
.
- If
is odd the form
is skew-symmetric:
.
4 Definition of signature
Let be a symmetric bilinear form on a free
-module.
Denote by
(
) the number of positive (negative) eigenvalues.
Note that since is symmetric, it is diagonalisable over the real numbers, so
(
) is the dimension of a maximal subspace on which the form is positive (negative) definite.
Then the signature of is defined to be
Tex syntax error
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
Tex syntax erroris defined to be the signature of the intersection form of
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
5 Equivalence of bilinear forms
Let and
be unimodular bilinear forms on underlying free
-modules
and
respectively. The forms
and
are called equivalent or isomorphic if there is an isomorphism
such that
.
The rank of is the rank of the underlying
-module
.
6 Skew-symmetric bilinear forms
The skew-symmetric hyperbolic form of rank ,
, is defined by the following intersection matrix
![\displaystyle \left( \begin{array}{cc} ~0 & ~1 \\ -1 & ~0 \end{array} \right) .](/images/math/f/3/a/f3ac0fdac865b4d550e21dcab0de6b1d.png)
Proposition 6.1.
Every skey-symmetri uni-modular bilinear form over ,
, isomorphic to the sum of some number of hyperbolic forms:
![\displaystyle q \cong \oplus_{i=1}^r H_-(\Zz).](/images/math/0/c/a/0ca6433f06c963c2693431c53b38e435.png)
In particular the rank of , in this case
, is even.
7 Symmetric bilinear forms
The classification of uni-modular definite symmetric bilinear forms is a deep and difficult problem. However the situation becomes much easier when the form is indefinite. Fundamental invariants are rank, signature and the following two.
A form is called definite if it is positive or negative definite, otherwise it is called indefinite.
A form may have two different types. It is of type even if
is an even number for any element
. Equivalently, if
is written as a square matrix in a basis, it is even if the elements on the diagonal are all even. Otherwise,
is said of type odd.
7.1 Classification of indefinite forms
There is a simple classification result of indefinite forms [Serre1970],[Milnor&Husemoller1973]:
![q, q'](/images/math/0/0/8/00867c957e16284b9243067ea7635467.png)
![\mathbb{Z}](/images/math/9/3/4/9349f2f6b15e7d4ce91c301ddd64bc06.png)
![q](/images/math/e/b/6/eb6af5b4e510c3c874d7d1f51d72393a.png)
![q'](/images/math/6/f/d/6fdb1bd0ad5208d967584a84fc65b257.png)
There is a further invariant of a unimodular symmetric bilinear form on
: An element
is called a characteristic vector of the form if one has
![\displaystyle q(c,x) \equiv q(x,x) \ (\text{mod} \ 2)](/images/math/a/7/4/a74ad4537107ba2c3a93c27b23682c57.png)
for all elements . Characteristic vectors always exist. In fact, when reduced modulo 2, the map
is linear. By unimodularity there therefore exists an element
such that the map
equals this linear map.
The form is even if and only if
is a characteristic vector.
If
and
are characteristic vectors for
, then there is an element
with
. This follows from unimodularity. As a consequence, the number
is independent of the chosen characteristic vector
modulo 8. One can be more specific:
Proposition 7.2.
For a characteristic vector of the unimodular symmetric bilinear form
one has
![\displaystyle q(c,c) \equiv \text{sign}(q) \ (\text{mod} \ 8)](/images/math/a/2/f/a2fb4177a192383dae312383b7f801c0.png)
Proof:
Suppose is a characteristic vector of
. Then
is a characteristic vector of the form
![\displaystyle q' = q \oplus \begin{pmatrix} 1 & \ 0 \\ 0 & -1 \end{pmatrix},](/images/math/4/3/a/43a487a575603d4bc0d597fc65ae01f3.png)
where form basis elements of the additional
summand with square
. We notice that
![\displaystyle q(c,c) = q'(c+e_+ + e_-, c+e_+ + e_-) .](/images/math/8/1/a/81a26702a8c96dd149c5023026f12b6c.png)
However, the form is indefinite, so the above classification theorem applies. In particular,
is odd and has the same signature as
, so it is equivalent to the diagonal form with
summands of (+1) and
summands of
. This diagonal form has a characteristic vector
that is simply a sum of basis elements in which the form is diagonal. Of course
. The claim now follows from the fact that the square of a characteristic vector is independent of the chosen characteristic vector modulo 8.
Corollary 7.3. The signature of an even (definite or indefinite) form is divisible by 8.
7.2 Examples, Realisations of indefinite forms
We shall show that any indefinite form permitted by the above theorem and corollary can be realised.
All possible values of rank and signature of odd forms are realised by direct sums of the forms of rank 1,
![\displaystyle b^+ (+1) \oplus b^- (-1) .](/images/math/d/1/b/d1bc34c5d3fda6e7a65a01c93b65e5f5.png)
An even positive definite form of rank 8 is given by the matrix
![\displaystyle E_8 = \left( \begin{array}{c c c c c c c c} \ 2 \ & \ 1\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ \\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 \\ \end{array} \right) .](/images/math/a/b/d/abd2d8d6026c5ebb8731ece7dcecc28c.png)
Likewise, the matrix represents a negative definite even form of rank 8.
Tex syntax errorgiven by
![\displaystyle H = \begin{pmatrix} \ 0 \ & \ 1 \ \\ 1 & 0 \end{pmatrix}](/images/math/3/c/5/3c5111cc3e76964de1e2d334965b144f.png)
determines an indefinite even form of rank 2 and signature 0. It is easy to see that the direct sums
![\displaystyle k E_8 \oplus l H](/images/math/3/9/4/394f2f02da4ec0f3793c0495ea6dc3a5.png)
with realise all unimodular symmetric indefinite even forms that are allowed by the above classification result. Here we use the convention that
is the
-fold direct sum of
for positive
and
is the
-fold direct sum of the negative definite form
.
8 References
- [Kirby1989] R.C. Kirby, The topology of 4-manifolds, Lecture Notes in Math. 1374, Springer-Verlag, 1989. MR1001966 (90j:57012)
- [Milnor&Husemoller1973] J. Milnor and D. Husemoller, Symmetric bilinear forms, Springer-Verlag, New York, 1973. MR0506372 (58 #22129) Zbl 0292.10016
- [Serre1970] J. Serre, Cours d'arithmétique, Presses Universitaires de France, Paris, 1970. MR0255476 (41 #138) Zbl 0432.10001
- [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
- [Skopenkov2015b] A. Skopenkov, Algebraic Topology From Geometric Viewpoint (in Russian), MCCME, Moscow, 2015, 2020. Accepted for English translation by `Moscow Lecture Notes' series of Springer. Preprint of a part
9 External links
- The Wikipedia page on Poincaré duality