This is a glossary of properties and concepts in algebraic topology in mathematics.

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Algebraic topology is to map topological spaces into groups (or other algebraic structures) in such a way that continuous functions between topological spaces map to homomorphisms between their associated groups.1 In Sections 2, 3 and 4, this chapter covers the three basic constructions of algebraic topology: homotopy, homology and cohomology. Algebraic Topology I. Course Home Syllabus Calendar Lecture Notes Assignments Download Course Materials; The Hopf fibration shows how the three-sphere can be built by a collection of circles arranged like points on a two-sphere. This is a frame from an animation of.

See also: glossary of topology, list of algebraic topology topics, glossary of category theory, glossary of differential geometry and topology, Timeline of manifolds.

• Convention: Throughout the article, I denotes the unit interval, Sⁿ the n-sphere and Dⁿ the n-disk. Also, throughout the article, spaces are assumed to be reasonable; this can be taken to mean for example, a space is a CW complex or compactly generatedweakly Hausdorff space. Similarly, no attempt is made to be definitive about the definition of a spectrum. A simplicial set is not thought of as a space; i.e., we generally distinguish between simplicial sets and their geometric realizations.
• Inclusion criterion: As there is no glossary of homological algebra in Wikipedia right now, this glossary also includes some few concepts in homological algebra (e.g., chain homotopy); some concepts in geometric topology are also fair game. On the other hand, the items that appear in glossary of topology are generally omitted. Abstract homotopy theory and motivic homotopy theory are also outside the scope. Glossary of category theory covers (or will cover) concepts in theory of model categories.

!$@[edit]

*

The base point of a based space.

${\displaystyle X_{+}}$

For an unbased space X, X₊ is the based space obtained by adjoining a disjoint base point.

A[edit]

absolute neighborhood retract

abstract

1. Abstract homotopy theory

Adams

1. John Frank Adams.

2. The Adams spectral sequence.

3. The Adams conjecture.

4. The Adams e-invariant.

5. The Adams operations.

Alexander duality

Alexander trick

The Alexander trick produces a section of the restriction map ${\displaystyle \mathrm{Top}(D^{n+1})\to \mathrm{Top}(S^n)}$, Top denoting a homeomorphism group; namely, the section is given by sending a homeomorphism ${\displaystyle f:S^n\to S^n}$ to the homeomorphism

${\displaystyle \tilde{f}:D^{n+1}\to D^{n+1},0\mapsto 0,0\ne x\mapsto xf(x/x)}$.

This section is in fact a homotopy inverse.^[1]

Analysis Situs

aspherical space

assembly map

Atiyah

1. Michael Atiyah.

2. Atiyah duality.

3. The Atiyah–Hirzebruch spectral sequence.

B[edit]

bar construction

based space

A pair (X, x₀) consisting of a space X and a point x₀ in X.

Betti number

Bockstein homomorphism

Borel–Moore homology

Borsuk's theorem

Bott

1. Raoul Bott.

2. The Bott periodicity theorem for unitary groups say: ${\displaystyle {\pi }_{q}U={\pi }_{q+2}U,q\ge 0}$.

3. The Bott periodicity theorem for orthogonal groups say: ${\displaystyle {\pi }_{q}O={\pi }_{q+8}O,q\ge 0}$.

Brouwer fixed point theorem

The Brouwer fixed-point theorem says that any map ${\displaystyle f:D^n\to D^n}$ has a fixed point.

C[edit]

cap product

Čech cohomology

cellular

1. A map ƒ:X→Y between CW complexes is cellular if ${\displaystyle f(X^n)\subset Y^n}$ for all n.

2. The cellular approximation theorem says that every map between CW complexes is homotopic to a cellular map between them.

3. The cellular homology is the (canonical) homology of a CW complex. Note it applies to CW complexes and not to spaces in general. A cellular homology is highly computable; it is especially useful for spaces with natural cell decompositions like projective spaces or Grassmannian.

chain homotopy

Given chain maps ${\displaystyle f,g:(C,d_C)\to (D,d_D)}$ between chain complexes of modules, a chain homotopys from f to g is a sequence of module homomorphisms ${\displaystyle s_i:C_i\to D_{i+1}}$ satisfying ${\displaystyle f_i-g_i=d_D\circ s_i+s_{i-1}\circ d_C}$.

chain map

A chain map ${\displaystyle f:(C,d_C)\to (D,d_D)}$ between chain complexes of modules is a sequence of module homomorphisms ${\displaystyle f_i:C_i\to D_i}$ that commutes with the differentials; i.e., ${\displaystyle d_D\circ f_i=f_{i-1}\circ d_C}$.

chain homotopy equivalence

A chain map that is an isomorphism up to chain homotopy; that is, if ƒ:C→D is a chain map, then it is a chain homotopy equivalence if there is a chain map g:D→C such that gƒ and ƒg are chain homotopic to the identity homomorphisms on C and D, respectively.

change of fiber

The change of fiber of a fibration p is a homotopy equivalence, up to homotopy, between the fibers of p induced by a path in the base.

character variety

The character variety^[2] of a group π and an algebraic group G (e.g., a reductive complex Lie group) is the geometric invariant theory quotient by G:

${\displaystyle \mathcal{X}(\pi ,G)=\mathrm{Hom}(\pi ,G)//G}$.

Munkres Algebraic Topology Pdf

D[edit]

deck transformation

Another term for an automorphism of a covering.

delooping

degeneracy cycle

degree

E[edit]

Eckmann–Hilton argument

The Eckmann–Hilton argument.

Eckmann–Hilton duality

Eilenberg–MacLane spaces

Given an abelian group π, the Eilenberg–MacLane spaces${\displaystyle K(\pi ,n)}$ are characterized by

.

F[edit]

factorization homology

fiber-homotopy equivalence

Given D→B, E→B, a map ƒ:D→E over B is a fiber-homotopy equivalence if it is invertible up to homotopy over B. The basic fact is that if D→B, E→B are fibrations, then a homotopy equivalence from D to E is a fiber-homotopy equivalence.

fibration

A map p:E → B is a fibration if for any given homotopy ${\displaystyle g_t:X\to B}$ and a map ${\displaystyle h_0:X\to E}$ such that ${\displaystyle p\circ h_0=g_0}$, there exists a homotopy ${\displaystyle h_t:X\to E}$ such that ${\displaystyle p\circ h_t=g_t}$. (The above property is called the homotopy lifting property.) A covering map is a basic example of a fibration.

fibration sequence

One says ${\displaystyle F\to X\stackrel{p}{\to }B}$ is a fibration sequence to mean that p is a fibration and that F is homotopy equivalent to the homotopy fiber of p, with some understanding of base points.

finitely dominated

fundamental class

fundamental group

The fundamental group of a space X with base point x₀ is the group of homotopy classes of loops at x₀. It is precisely the first homotopy group of (X, x₀) and is thus denoted by ${\displaystyle {\pi }_1(X,x_0)}$.

fundamental groupoid

The fundamental groupoid of a space X is the category whose objects are the points of X and whose morphisms x → y are the homotopy classes of paths from x to y; thus, the set of all morphisms from an object x₀ to itself is, by definition, the fundament group ${\displaystyle {\pi }_1(X,x_0)}$.

free

Synonymous with unbased. For example, the free path space of a space X refers to the space of all maps from I to X; i.e., ${\displaystyle X^I}$ while the path space of a based space X consists of such map that preserve the base point (i.e., 0 goes to the base point of X).

Freudenthal suspension theorem

For a nondegenerately based space X, the Freudenthal suspension theorem says: if X is (n-1)-connected, then the suspension homomorphism

${\displaystyle {\pi }_{q}X\to {\pi }_{q+1}\mathrm{\Sigma }X}$

is bijective for q < 2n - 1 and is surjective if q = 2n - 1.

G[edit]

G-fibration

A G-fibration with some topological monoidG. An example is Moore's path space fibration.

Γ-space

generalized cohomology theory

A generalized cohomology theory is a contravariant functor from the category of pairs of spaces to the category of abelian groups that satisfies all of the Eilenberg–Steenrod axioms except the dimension axiom.

genus

group completion

grouplike

An H-space X is said to be group-like or grouplike if ${\displaystyle {\pi }_{0}X}$ is a group; i.e., X satisfies the group axioms up to homotopy.

Gysin sequence

H[edit]

Hilton–Milnor theorem

The Hilton–Milnor theorem.

H-space

An H-space is a based space that is a unital magma up to homotopy.

homologus

Two cycles are homologus if they belong to the same homology class.

homotopy category

Let C be a subcategory of the category of all spaces. Then the homotopy category of C is the category whose class of objects is the same as the class of objects of C but the set of morphisms from an object x to an object y is the set of the homotopy classes of morphisms from x to y in C. For example, a map is a homotopy equivalence if and only if it is an isomorphism in the homotopy category.

homotopy colimit

homotopy over a space B

A homotopy h_t such that for each fixed t, h_t is a map over B.

homotopy equivalence

1. A map ƒ:X→Y is a homotopy equivalence if it is invertible up to homotopy; that is, there exists a map g: Y→X such that g ∘ ƒ is homotopic to th identity map on X and ƒ ∘ g is homotopic to the identity map on Y.

2. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between the two. For example, by definition, a space is contractible if it is homotopy equivalent to a point space.

homotopy excision theorem

The homotopy excision theorem is a substitute for the failure of excision for homotopy groups.

homotopy fiber

The homotopy fiber of a based map ƒ:X→Y, denoted by Fƒ, is the pullback of ${\displaystyle PY\to Y,\chi \mapsto \chi (1)}$ along f.

homotopy fiber product

A fiber product is a particular kind of a limit. Replacing this limit lim with a homotopy limit holim yields a homotopy fiber product.

homotopy group

1. For a based space X, let ${\displaystyle {\pi }_{n}X=[S^n,X]}$, the set of homotopy classes of based maps. Then ${\displaystyle {\pi }_{0}X}$ is the set of path-connected components of X, ${\displaystyle {\pi }_{1}X}$ is the fundamental group of X and ${\displaystyle {\pi }_{n}X,n\ge 2}$ are the (higher) n-th homotopy groups of X.

2. For based spaces ${\displaystyle A\subset X}$, the relative homotopy group${\displaystyle {\pi }_n(X,A)}$ is defined as ${\displaystyle {\pi }_{n-1}}$ of the space of paths that all start at the base point of X and end somewhere in A. Equivalently, it is the ${\displaystyle {\pi }_{n-1}}$ of the homotopy fiber of ${\displaystyle A\hookrightarrow X}$.

3. If E is a spectrum, then ${\displaystyle {\pi }_{k}E=\underset{n}{\underrightarrow{\lim }}{\pi }_{k+n}{E}_n.}$

4. If X is a based space, then the stable k-th homotopy group of X is ${\displaystyle {\pi }_k^{s}X=\underset{n}{\underrightarrow{\lim }}{\pi }_{k+n}{\mathrm{\Sigma }}^{n}X}$. In other words, it is the k-th homotopy group of the suspension spectrum of X.

homotopy quotient

If G is a Lie group acting on a manifold X, then the quotient space ${\displaystyle (EG\times X)/G}$ is called the homotopy quotient (or Borel construction) of X by G, where EG is the universal bundle of G.

homotopy spectral sequence

homotopy sphere

Hopf

1. Heinz Hopf.

2. Hopf invariant.

3. The Hopf index theorem.

4. Hopf construction.

Hurewicz

The Hurewicz theorem establishes a relationship between homotopy groups and homology groups.

I[edit]

infinite loop space

infinite loop space machine

infinite mapping telescope

integration along the fiber

isotopy

J[edit]

J-homomorphism

See J-homomorphism.

join

The join of based spaces X, Y is ${\displaystyle X\star Y=\mathrm{\Sigma }(X\wedge Y).}$

K[edit]

k-invariant

Kan complex

See Kan complex.

Kervaire invariant

The Kervaire invariant.

Koszul duality

Koszul duality.

Künneth formula

L[edit]

Lazard ring

The Lazard ringL is the (huge) commutative ring together with the formal group law ƒ that is universal among all the formal group laws in the sense that any formal group law g over a commutative ring R is obtained via a ring homomorphism L → R mapping ƒ to g. According to Quillen's theorem, it is also the coefficient ring of the complex bordism MU. The Spec of L is called the moduli space of formal group laws.

Lefschetz fixed point theorem

The Lefschetz fixed point theorem says: given a finite simplicial complex K and its geometric realization X, if a map ${\displaystyle f:X\to X}$ has no fixed point, then the Lefschetz number of f; that is,

${\displaystyle \sum _0^{\mathrm{\infty }}(-1{)}^q\mathrm{tr}(f_{\ast }:{\mathrm{H}}_q(X)\to {\mathrm{H}}_q(X))}$

is zero. For example, it implies the Brouwer fixed-point theorem since the Lefschetz number of ${\displaystyle f:D^n\to D^n}$ is, as higher homologies vanish, one.

lens space

The lens space is the quotient space ${\displaystyle \{z\in {\mathbb{C}}^{n}z=1\}/{\mu }_p}$ where ${\displaystyle {\mu }_p}$ is the group of p-th roots of unity acting on the unit sphere by ${\displaystyle \zeta \cdot (z_1,\dots ,z_n)=(\zeta z_1,\dots ,\zeta z_n)}$.

Leray spectral sequence

local coefficient

1. A module over the group ring${\displaystyle \mathbb{Z}[{\pi }_{1}B]}$ for some based space B; in other words, an abelian group together with a homomorphism ${\displaystyle {\pi }_{1}B\to \mathrm{Aut}(A)}$.

2. The local coefficient system over a based space B with an abelian group A is a fiber bundle over B with discrete fiber A. If B admits a universal covering ${\displaystyle \tilde{B}}$, then this meaning coincides with that of 1. in the sense: every local coefficient system over B can be given as the associated bundle${\displaystyle \tilde{B}{\times }_{{\pi }_{1}B}A}$.

local sphere

The localization of a sphere at some prime number

localization

locally constant sheaf

A locally constant sheaf on a space X is a sheaf such that each point of X has an open neighborhood on which the sheaf is constant.

loop space

The loop space${\displaystyle \mathrm{\Omega }X}$ of a based space X is the space of all loops starting and ending at the base point of X.

M[edit]

N[edit]

n-cell

Another term for an n-disk.

n-connected

A based space X is n-connected if ${\displaystyle {\pi }_{q}X=0}$ for all integers q ≤ n. For example, '1-connected' is the same thing as 'simply connected'.

n-equivalent

NDR-pair

A pair of spaces ${\displaystyle A\subset X}$ is said to be an NDR-pair (=neighborhood deformation retract pair) if there is a map ${\displaystyle u:X\to I}$ and a homotopy ${\displaystyle h_t:X\to X}$ such that ${\displaystyle A=u^{-1}(0)}$, ${\displaystyle h_0={\mathrm{id}}_X}$, ${\displaystyle h_t{}_A={\mathrm{id}}_A}$ and .
If A is a closed subspace of X, then the pair ${\displaystyle A\subset X}$ is an NDR-pair if and only if ${\displaystyle A\hookrightarrow X}$ is a cofibration.

nilpotent

1. nilpotent space; for example, a simply connected space is nilpotent.

2. The nilpotent theorem.

nonabelian

1. nonabelian cohomology

2. nonabelian algebraic topology

normalized

Given a simplicial groupG, the normalized chain complexNG of G is given by ${\displaystyle (NG{)}_n={\cap }_1^{\mathrm{\infty }}\ker d_i^n}$ with the n-th differential given by ${\displaystyle d_0^n}$; intuitively, one throws out degenerate chains.^[5] It is also called the Moore complex.

O[edit]

obstruction cocycle

obstruction theory

Obstruction theory is the collection of constructions and calculations indicating when some map on a submanifold (subcomplex) can or cannot be extended to the full manifold. These typically involve the Postnikov tower, killing homotopy groups, obstruction cocycles, etc.

of finite type

A CW complex is of finite type if there are only finitely many cells in each dimension.

operad

The portmanteau of “operations” and “monad”. See operad.

orbit category

orientation

1. The orientation covering (or orientation double cover) of a manifold is a two-sheeted covering so that each fiber over x corresponds to two different ways of orienting a neighborhood of x.

2. An orientation of a manifold is a section of an orientation covering; i.e., a consistent choice of a point in each fiber.

3. An orientation character (also called the first Stiefel–Whitney class) is a group homomorphism ${\displaystyle {\pi }_1(X,x_0)\to \{\pm 1\}}$ that corresponds to an orientation covering of a manifold X (cf. #covering.)

4. See also orientation of a vector bundle as well as orientation sheaf.

P[edit]

p-adic homotopy theory

The p-adic homotopy theory.

path class

An equivalence class of paths (two paths are equivalent if they are homotopic to each other).

path lifting

A path lifting function for a map p: E → B is a section of ${\displaystyle E^I\to P_p}$ where ${\displaystyle P_p}$ is the mapping path space of p. For example, a covering is a fibration with a unique path lifting function. By formal consideration, a map is a fibration if and only if there is a path lifting function for it.

path space

The path space of a based space X is ${\displaystyle PX=\mathrm{Map}(I,X)}$, the space of based maps, where the base point of I is 0. Put in another way, it is the (set-theoretic) fiber of ${\displaystyle X^I\to X,\chi \mapsto \chi (0)}$ over the base point of X. The projection ${\displaystyle PX\to X,\chi \mapsto \chi (1)}$ is called the path space fibration, whose fiber over the base point of X is the loop space ${\displaystyle \mathrm{\Omega }X}$. See also mapping path space.

phantom map

Poincaré

The Poincaré duality theorem says: given a manifold M of dimension n and an abelian group A, there is a natural isomorphism

${\displaystyle {\mathrm{H}}_c^{\ast }(M;A)\simeq {\mathrm{H}}_{n-\ast }(M;A)}$.

Q[edit]

quasi-fibration

A quasi-fibration is a map such that the fibers are homotopy equivalent to each other.

Quillen

1. Daniel Quillen

2. Quillen’s theorem says that ${\displaystyle {\pi }_{\ast }MU}$ is the Lazard ring.

R[edit]

rational

1. The rational homotopy theory.

2. The rationalization of a space X is, roughly, the localization of X at zero. More precisely, X₀ together with j: X → X₀ is a rationalization of X if the map ${\displaystyle {\pi }_{\ast }X\otimes \mathbb{Q}\to {\pi }_{\ast }{X}_0\otimes \mathbb{Q}}$ induced by j is an isomorphism of vector spaces and ${\displaystyle {\pi }_{\ast }{X}_0\otimes \mathbb{Q}\simeq {\pi }_{\ast }{X}_0}$.

3. The rational homotopy type of X is the weak homotopy type of X₀.

Reidemeister

Reidemeister torsion.

reduced

The reduced suspension of a based space X is the smash product ${\displaystyle \mathrm{\Sigma }X=X\wedge S^1}$. It is related to the loop functor by ${\displaystyle \mathrm{Map}(\mathrm{\Sigma }X,Y)=\mathrm{Map}(X,\mathrm{\Omega }Y)}$ where ${\displaystyle \mathrm{\Omega }Y=\mathrm{Map}(S^1,Y)}$ is the loop space.

ring spectrum

A ring spectrum is a spectrum that satisfying the ring axioms, either on nose or up to homotopy. For example, a complex K-theory is a ring spectrum.

S[edit]

Samelson product

Serre

1. Jean-Pierre Serre.

2. Serre class.

3. Serre spectral sequence.

simple

simple-homotopy equivalence

A map ƒ:X→Y between finite simplicial complexes (e.g., manifolds) is a simple-homotopy equivalence if it is homotopic to a composition of finitely many elementary expansions and elementary collapses. A homotopy equivalence is a simple-homotopy equivalence if and only if its Whitehead torsion vanishes.

simplicial approximation

See simplicial approximation theorem.

simplicial complex

See simplicial complex; the basic example is a triangulation of a manifold.

simplicial homology

A simplicial homology is the (canonical) homology of a simplicial complex. Note it applies to simplicial complexes and not to spaces; cf. #singular homology.

signature invariant

singular

1. Given a space X and an abelian group π, the singular homology group of X with coefficients in π is

${\displaystyle {\mathrm{H}}_{\ast }(X;\pi )={\mathrm{H}}_{\ast }(C_{\ast }(X)\otimes \pi )}$

where ${\displaystyle C_{\ast }(X)}$ is the singular chain complex of X; i.e., the n-th degree piece is the free abelian group generated by all the maps ${\displaystyle {\mathrm{△}}^n\to X}$ from the standard n-simplex to X. A singular homology is a special case of a simplicial homology; indeed, for each space X, there is the singular simplicial complex of X^[6] whose homology is the singular homology of X.

2. The singular simplices functor is the functor ${\displaystyle \mathbf{T}\mathbf{o}\mathbf{p}\to s\mathbf{S}\mathbf{e}\mathbf{t}}$ from the category of all spaces to the category of simplicial sets, that is the right adjoint to the geometric realization functor.

3. The singular simplicial complex of a space X is the normalized chain complex of the singular simplex of X.

slant product

small object argument

smash product

The smash product of based spaces X, Y is ${\displaystyle X\wedge Y=X\times Y/X\vee Y}$. It is characterized by the adjoint relation

${\displaystyle \mathrm{Map}(X\wedge Y,Z)=\mathrm{Map}(X,\mathrm{Map}(Y,Z))}$.

Allen Hatcher Algebraic Topology Pdf

T[edit]

Thom

1. René Thom.

2. If E is a vector bundle on a paracompact space X, then the Thom space${\displaystyle \text{Th}(E)}$ of E is obtained by first replacing each fiber by its compactification and then collapsing the base X.

3. The Thom isomorphism says: for each orientable vector bundleE of rank n on a manifold X, a choice of an orientation (the Thom class of E) induces an isomorphism

${\displaystyle {\tilde{\mathrm{H}}}^{\ast +n}(\text{Th}(E);\mathbb{Z})\simeq {\mathrm{H}}^{\ast }(X;\mathbb{Z})}$.

Topology Mathematics Pdf

U[edit]

universal coefficient

The universal coefficient theorem.

up to homotopy

A statement holds in the homotopy category as opposed to the category of spaces.

V[edit]

Algebraic Topology Hatcher Pdf

van Kampen

The van Kampen theorem says: if a space X is path-connected and if x₀ is a point in X, then

${\displaystyle {\pi }_1(X,x_0)=\underrightarrow{\lim }{\pi }_1(U,x_0)}$

where the colimit runs over some open cover of X consisting of path-connected open subsets containing x₀ such that the cover is closed under finite intersections.

Bredon Algebraic Topology Pdf

W[edit]

Switzer Algebraic Topology Pdf

Waldhausen S-construction

Waldhausen S-construction.

Wall's finiteness obstruction

weak equivalence

A map ƒ:X→Y of based spaces is a weak equivalence if for each q, the induced map ${\displaystyle f_{\ast }:{\pi }_{q}X\to {\pi }_{q}Y}$ is bijective.

wedge

For based spaces X, Y, the wedge product${\displaystyle X\wedge Y}$ of X and Y is the coproduct of X and Y; concretely, it is obtained by taking their disjoint union and then identifying the respective base points.

well pointed

A based space is well pointed (or non-degenerately based) if the inclusion of the base point is a cofibration.

Whitehead

1. J. H. C. Whitehead.

2. Whitehead's theorem says that for CW complexes, the homotopy equivalence is the same thing as the weak equivalence.

3. Whitehead group.

4. Whitehead product.

winding number

Notes[edit]

1. ^Let r, s denote the restriction and the section. For each f in ${\displaystyle \mathrm{Top}(D^{n+1})}$, define ${\displaystyle h_t(f)(x)=tf(x/t),x\le t,h_t(f)(x)=xf(x/x),x>t}$. Then ${\displaystyle h_t:s\circ r\sim \mathrm{id}}$.
2. ^Despite the name, it may not be an algebraic variety in the strict sense; for example, it may not be irreducible. Also, without some finiteness assumption on G, it is only a scheme.
3. ^Hatcher, Ch. 4. H.
4. ^How to think about model categories?
5. ^https://ncatlab.org/nlab/show/Moore+complex
6. ^http://ncatlab.org/nlab/show/singular+simplicial+complex

References[edit]

• J.F. Adams, Stable Homotopy and Generalized Homology, Chicago Lectures in Mathematics, The University of Chicago Press, 1974.
• Adams, J. (1978), Infinite loop spaces, Princeton University Press, Princeton, New Jersey.
• Bott, Raoul; Tu, Loring (1982), Differential Forms in Algebraic Topology, New York: Springer, ISBN0-387-90613-4
• Bousfield, A. K.; Kan, D. M. (1987), Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics, 304, Springer, ISBN9783540061052
• James F. Davis, Paul Kirk, Lecture Notes in Algebraic Topology
• Fulton, William, Algebraic Topology
• Allen Hatcher, Algebraic topology.
• Hess, Kathryn (2006-04-28). 'Rational homotopy theory: a brief introduction'. arXiv:math/0604626. Bibcode:2006math...4626H.Cite journal requires journal= (help)
• algebraic topology, Lectures delivered by Michael Hopkins and Notes by Akhil Mathew, Fall 2010, Harvard
• Lurie, J. Algebraic K-Theory and Manifold Topology (Math 281)
• Lurie, J. Chromatic Homotopy Theory (252x)
• May, J. A Concise Course in Algebraic Topology
• May, J. (with K. Ponto) More concise algebraic topology: localization, completion, and model categories
• May and Sigurdsson, Parametrized homotopy theory (despite the title, it contains a significant amount of general results.)
• Dennis Sullivan, Geometric Topology, the 1970 MIT notes
• George William Whitehead (1978). Elements of homotopy theory. Graduate Texts in Mathematics. 61 (3rd ed.). New York-Berlin: Springer-Verlag. pp. xxi+744. ISBN978-0-387-90336-1. MR0516508. Retrieved September 6, 2011.
• Kirsten Graham Wickelgren, 8803 Stable Homotopy Theory

Munkres Elements Of Algebraic Topology Pdf

External links[edit]

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